Prof. Dr. Dr. hc G

CLASSIFICATION OF SINGULARITIES IN POSITIVE CHARACTRISTIC

NGUYEN HONG DUC

Vom Fachbereich Mathematik der Universität Kaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften (Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation.

1. Gutachter: Prof. Dr. Dr. h.c. G.-M. Greuel 2. Gutachter: Prof. Dr. A. Campillo Vollzug der Promotion: 18.04.2013

D 386

To my family to my wife Tran Minh and to our daughter Ngan Ha

Introduction In this thesis we treat the classification hypersurface singularities in K[[x1 , . . . , xn ]], K an algebraically closed field of characteristic p > 0 w.r.t. right resp. contact equivalence. In [Arn72], [Arn73], [Arn76] Arnol’d classified simple, unimodal and bimodal singularities for K = R and C w.r.t. right equivalence. He showed that the simple singularities are exactly the ADE-singularities, i.e. the two infinite series Ak , k ≥ 1, Dk , k ≥ 4, and the three exceptional singularities E6 , E7 , E8 . It turned out later by Giusti in [Giu77] that the ADE-singularities of Arnol’d are also exactly those of modality 0 for contact equivalence. The classification of contact unimodal singularities was achieved in [Wal83] for K = C. In the late eighties, Greuel and Kröning showed in [GK90] that the contact simple singularities over a field of positive characteristic are again exactly the ADE-singularities but with a few more normal forms in small characteristic. While the classification of singularities w.r.t. right equivalence and higher modality is still missing in positive characteristic, some partial results were obtained in [DG98], [BGM11]. Based on these works we start to classify singularities in positive characteristic. We completely classify right simple singularities. It is surprising that w.r.t. right equivalence and any given p > 0 we have only finitely many simple singularities, i.e. there are only finitely many k such that Ak and Dk are right simple, all the others have moduli. A complete right classification of all univariate power series is achieved by explicit normal forms. The classification of unimodal singularities w.r.t. contact equivalence appears to be rather involved. In this thesis we start the classification of contact unimodal singularities by giving pre-normal forms. The plane curve singularities in positive characteristic are discussed in more detail. We compute and relate (classical and new) numerical invariants for plane curve singularities. Now we give a more detailed description of the content. The thesis consists of five chapters: • Chapter 1 contains preliminaries and we fix the notations. • Chapter 2 deals with invariants of plane curve singularities. • Chapter 3 studies the notion of modality and its relation to deformation theory. • Chapter 4 contains a complete right classification of simple singularities and all univariate power series in positive characteristic. • Chapter 5 is the first step to classify contact unimodal singularities. 1

2 In the first two sections of Chapter 2, we consider different notions of nondegeneracy, as introduced by Kouchnirenko (NND), Wall (INND) and BeelenPellikaan (WNND) for plane curve singularities {f (x, y) = 0} and introduce the new notion of weighted homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number µ resp. the delta-invariant δ can be computed by explicit formulas µN resp. δN from the Newton diagram of f if f is NND (or INND) resp. WNND. It was however unknown whether the equalities µ = µN resp. δ = δN can be characterized by a certain non-degeneracy condition on f and, if so, by which one. We show that µ = µN resp. δ = δN is equivalent to INND resp. WHNND. The next two sections (2.3, 2.4) are motivated by the Milnor formula [Mil68]: If f ∈ K[[x, y]] with char(K) = 0 then µ(f ) = 2δ(f ) − r(f ) + 1, where r(f ) denotes the number of branches of f . It is known that this is wrong in positive characteristic and the difference µ(f ) − (2δ(f ) − r(f ) + 1) = Sw(f ) ≥ 0, where Sw(f ) denotes the Swan character, which measures wild vanishing cycles (of the Milnor fiber) of f . Section 2.3 is to approach the following question: How to characterize the plane curve singularities without wild vanishing cycles if the characteristic p is positive? We introduce in Section 2.3 a new invariant (Gamma-invariant, γ), a notion of im-goodness of p for f (“im” refers to intersection multiplicity), and prove that one has always (Proposition 2.3.7) γ(f ) ≥ 2δ(f ) − r(f ) + 1 with equality if p is im-good for f . We introduce a notion of m-goodness of p for f (“m” refers to multiplicity) and compute the Swan character of f by a polar characteristic difference (pcd(f )) if p is m-good for f (Theorem 2.4.4). In the final section of this chapter (Section 2.5) we prove that every reduced plane curve singularity f is finitely parametrization determined. Moreover we show that the parametrization determinacy of f is at most c + 1, where c is the conductor (exponent) of f (Theorem 2.5.3). In Chapter 3 we give a precise definition of the number of moduli (modality) for families of power series parametrized by an algebraic variety. In fact, we give two definitions of G-modality, both related to the action of an algebraic group G on a variety X and show that they coincide (Propositions 3.3.3), a result which is valid in any characteristic. Moreover, we prove that the G-modality is upper semicontinuous for G the right resp. the contact group (Proposition 3.4.7). We introduce the notion of G-completeness for algebraic families (not just for germs) which suffices to determine the modality and generalize the Kas-Schlessinger theorem [KaS72] to deformations (unfoldings) of formal power series over algebraic

3 varieties. We prove that an algebraic representative of the semiuniversal deformation with section of an isolated hypersurface singularity is G-complete (for G the right resp. the contact group, see Proposition 3.4.16). However, in contrast to the complex analytic case the usual semiuniversal deformation is not sufficient to determine the modality and hence is not G-complete; we have to consider versal deformations with section (cf. Example 3.4.15). In Section 4.2 of Chapter 4 we give a normal form for any univariate power series f w.r.t. right equivalence (Proposition. 4.2.7). We show that the right modality of f is equal to the integer part of µ(f )/p (Theorem 4.2.8). As a consequence we prove that the right modality is equal to the dimension of the µ-constant stratum in an algebraic representative of the semiuniversal deformation with trivial section (Corollary 4.2.10). In Section 4.3 we classify completely right simple singularities in any dimension with tables of normal forms (Theorems 4.3.1-4.3.3). A surprising fact of our classification is that for any fixed p > 0 there exist only finitely many right simple singularities. For example, if p = 2 and n is even, there is just one right simple hypersurface, x1 x2 + x3 x4 + . . . + xn−1 xn , while for n odd no right simple singularity exists. We revisit in Chapter 5 the Drozd-Greuel ideal-unimodal plane curve singularities. Drozd and Greuel in [DG98] introduced the notion of ideal-unimodal plane curve singularities (IUS) by a table of types E, T, W, Z , which are exactly the contact unimodal singularities in characteristic zero (Corollary 5.1.11). We first prove that the list of IUS in the table is not disjoint (Proposition 5.2.1), for example E2,2q−1 = T2q+3,3,2 , E2,2(l−1) = T3,2l+2,2 , . . .. As a first step to determine the contact modality of all IUS and then classify contact unimodal singularities, we give pre-normal forms of all IUS. This is done by using the theorem on parametrization finite determinacy (Theorem 2.5.3). Finally let us mention that Sections 2.1 and 2.2 have been published as [GN11], and that Chapter 3 and Section 4.3 have been submitted as [GN12]. In this thesis we provide in addition detailed proofs. Section 4.2 has been submitted as [Ng12], while Sections 2.3-2.5 have not been published or submitted so far.

Acknowledgment First and foremost, I am deeply indebted to my advisor Professor Gert-Martin Greuel for his continued enthusiasm, introducing me the subjects and teaching me many basic knowledge related to my thesis. Although he was very busy, he always found time to listen and to provide guidance and encouragement. I would like to express my deep gratitude to Professor Ha Huy Vui for introducing me to Singularity theory and supervising for my master thesis at Hanoi Institute of Mathematics. I gratefully acknowledge the help, valuable advice, discussion and support of my many colleagues at the working group Algebra, Geometrie und Computeralgebra (AGAG), especially Andreas Gathmann, Gerhard Pfister and Thomas Markwig with

4 their useful lectures and seminars in Algebraic Geometry. I would like to show my appreciation to the secretary of our group Petra Bäsell. She was always willing to help me and my colleagues. I would like to thank the Department of Mathematics, especially the Graduate School “Mathematics as a Key Technology”, and the International School for Graduate Studies (ISGS) for their great assistance given to me during my study period in Kaiserslautern. The financial support of the Deutscher Akademischer Austausch Dienst (DAAD) through the research program Mathematics in Industry and Commerce (MIC) at the University of Kaiserslautern is gratefully acknowledged. I also would like to thank my parents, my brother and my sisters for their infinite love, given to me the best education. Last but not least, I would especially like to thank my wife Tran Minh and our daughter Ngan Ha for their undivided love, understanding, continued encouragement and patience throughout all these long years.

Contents 1 Preliminaries 1.1 Hypersurface singularities . . . . . . . . . . . . . . . . . . . . . 1.1.1 Preliminary concepts . . . . . . . . . . . . . . . . . . . . 1.1.2 Isolated hypersurface singularities and finite determinacy 1.1.3 Newton nondegeneracy . . . . . . . . . . . . . . . . . . . 1.2 Plane curve singularities . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Parametrization . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Parametrization equivalence . . . . . . . . . . . . . . . . 1.2.3 Intersection multiplicity and invariants . . . . . . . . . . 1.2.4 Factorizations of plane curve singularities . . . . . . . . . 1.2.5 Resolution of plane curve singularities . . . . . . . . . . 2 Plane Curve Singularities 2.1 Milnor number . . . . . . . . 2.2 Delta-invariant . . . . . . . . 2.3 Gamma-invariant . . . . . . . 2.4 Milnor formula . . . . . . . . 2.5 Parametrization determinacy .

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3 Modality 3.1 The fibers of a morphism . . . . . . . . 3.2 G-modality . . . . . . . . . . . . . . . 3.3 G-modality with respect to a morphism 3.4 Right and contact modality . . . . . .

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4 Right Classification of Singularities 4.1 Contact simple hypersurface singularities . . . . . . . . . . . . . . . 4.2 Right classification of univariate power series . . . . . . . . . . . . . 4.2.1 Normal forms of univariate power series . . . . . . . . . . . . 4.2.2 Right modality . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Right simple singularities . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Right simple plane curve singularities in characteristic > 2 . 4.3.2 Right simple hypersurface singularities in characteristic > 2 4.3.3 Right simple hypersurface singularities in characteristic 2 . . 5

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29 29 37 47 58 60

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5 Drozd-Greuel’s Ideal-Unimodal 5.1 Definition . . . . . . . . . . . 5.2 A disjoint list of IUS . . . . . 5.3 Pre-normal form . . . . . . . Bibliography

Plane Curve Singularities 113 . . . . . . . . . . . . . . . . . . . . . . 113 . . . . . . . . . . . . . . . . . . . . . . 116 . . . . . . . . . . . . . . . . . . . . . . 120

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Chapter 1 Preliminaries 1.1

Hypersurface singularities

Following Boubakri, Greuel and Markwig [BGM12] we review some of the standard facts on algebroid hypersurface singularities.

1.1.1

Preliminary concepts

Let K be an algebraically closed field, K[[x]] = K[[x1 , . . . , xn ]], n ≥ 1, the formal power series ring and m its maximal ideal. An (algebroid) singularity is a local K-algebra R which is isomorphic to K[[x]]/I, where I is a proper ideal of K[[x]]. If I = hf i, with f ∈ m \ {0} is a formal power series, then R is called an (algebroid) hypersurface singularity. Definition 1.1.1. Let f ∈ m \ {0} be a formal power series. 1. The ideal j(f ) := hfx1 , . . . , fxn i ⊂ K[[x]] is called the Jacobian ideal, or the Milnor ideal of f , and tj(f ) := hf i + j(f ) ⊂ K[[x]] is called the Tjurina ideal of f . Here fxi denotes the partial derivative of f with respect to xi . 2. The K-algebras Mf := K[[x]]/j(f ), Tf := K[[x]]/tj(f ) are called the Milnor and Tjurina algebra of f , respectively. 3. The numbers µ(f ) := dimK (Mf ), τ (f ) := dimK (Tf ) are called the Milnor and Tjurina numbers of f , respectively. 7

8

Chapter 1. Preliminaries

Let f ∈ m ⊂ K[[x]] be a non-zero element. By definition, if µ(f ) is finite, then so is τ (f ). If K = C, it is well-known that µ(f ) < ∞ ⇔ τ (f ) < ∞ (cf. [GLS06, Lemma 2.3]). Also, this statement is true also in characteristic zero (Theorem 1.1.5). However, in positive characteristic, this is in general not true as the following example shows. Let char(K) = 3 and let f (x, y) = x2 + y 3 ∈ K[[x, y]]. Then τ (f ) = 3 < µ(f ) = ∞. Remark 1.1.2. The functions µ, τ are upper semi-continuous. Proof. cf. Lemma 3.4.3. Definition 1.1.3. Let f, g ∈ m ⊂ K[[x]]. 1. f is called right equivalent to g, f ∼r g, if there exists an automorphism Φ of K[[x]] such that g = Φ(f ). 2. f is called contact equivalent to g, f ∼c g, if there exists an automorphism Φ of K[[x]] and a unit u ∈ K[[x]]∗ such that g = u · Φ(f ). It is straightforward from the above definition that the right and the contact equivalence are equivalence relations on the set of formal power series. Moreover it is known that f ∼r g implies f ∼c g, but the converse does not hold even in characteristic zero. Proposition 1.1.4. Let f, g ∈ m ⊂ K[[x]]. Furthermore, let Φ ∈ Aut(K[[x]]) be an automorphism of K[[x]] and let u ∈ (K[[x]])∗ be a unit. Then 1. j(Φ(f )) = Φ(j(f )). 2. huf i + j(uf ) = hf i + j(f ), or shortly tj(uf ) = tj(f ). 3. f ∼r g implies that Mf ∼ = Mg and Tf ∼ = Tg as K-algebras. In particular, µ(f ) = µ(g) and τ (f ) = τ (g). 4. f ∼c g implies that Tf ∼ = Tg and hence τ (f ) = τ (g). Proof. cf. [Bou09, Lemma 1.2.7] The above proposition says that the Milnor number is invariant under right equivalence and the Tjurina number is invariant under contact equivalence. In characteristic zero, Milnor number is even invariant under contact equivalence. Theorem 1.1.5 (Boubakri, Greuel and Markwig). Let K be an algebraically closed field of characteristic zero and f, g ∈ K[[x]]. 1. If f ∼c g, then µ(f ) = µ(g). 2. µ(f ) < ∞ if and only if τ (f ) < ∞. Proof. cf. [BGM12, Thm. 1, 2]

1.1 Hypersurface singularities

1.1.2

9

Isolated hypersurface singularities and finite determinacy

Definition 1.1.6. Let f ∈ m and let Rf = K[[x]]/hf i the associated hypersurface singularity. 1. f is called an isolated singularity, if there exists a k > 0 such that mk ⊂ j(f ). 2. Rf is called an isolated hypersurface singularity, if there exists a k > 0 such that mk ⊂ tj(f ). It is straightforward from the above definition that f is an isolated singularity (resp. Rf is an isolated hypersurface singularity) if and only if µ(f ) < ∞ ( resp. τ (f ) < ∞). Definition 1.1.7. Let k > 0 be a natural number. 1. We define the k-jet space by Jk := K[[x]]/mk+1 . 2. Let f =

P

α

aα xα be given. The k-jet of f is defined by j k (f ) =

X

aα x α .

|α|≤k

Definition 1.1.8. Let f ∈ K[[x]] and let k > 0 be a natural number. 1. f is called contact k-determined (resp. right k-determined) if all g ∈ K[[x]] with j k (g) = j k (f ) are contact equivalent (resp. right equivalent) to f . 2. If f is contact k-determined (resp. right k-determined) for some natural number k, then f is called contact finitely determined (resp. right finitely determined). The minimum such k is called the contact determinacy (resp. right determinacy) of f . It follows from Corollary 1.1.11 and Theorem 1.1.12 that “isolated singularity” resp. “isolated hypersurface singularity” is equivalent to “right finitely determined” resp. “contact finitely determined” Definition 1.1.9. Let f ∈ K[[x]] = K[[x1 , . . . , xn ]]. 1. We define the multiplicity of f , mt(f ), to be the largest integer k such that f ⊂ mk , and we set mt(0) = ∞. The multiplicity of f is also called the order of f and denoted by ord(f ).

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Chapter 1. Preliminaries 2. The differential order of f is defined by d(f ) := min{mt(fx1 ), . . . , mt(fxn )}.

Note that d(f ) = mt(f ) − 1 if char(K) - mt(f ), e.g. if char(K) = 0. Besides, d(f ) is invariant under right equivalence, while mt(f ) is even invariant under contact equivalence. Theorem 1.1.10 (Boubakri, Greuel and Markwig). Let 0 6= f ∈ m and let k > 0 be a natural number. 1. If mk+2 ⊂ m2 · j(f ), then f is right (2k − d(f ) + 1)-determined. 2. If mk+2 ⊂ m · hf i + m2 · j(f ), then f is contact (2k − mt(f ) + 2)-determined. Proof. cf. [BGM12, Thm. 3] Corollary 1.1.11 (Boubakri, Greuel and Markwig). Let 0 6= f ∈ m2 . 1. If µ(f ) < ∞, then the right determinacy of f is at most 2µ(f ) − d(f ) + 1. 2. If τ (f ) < ∞, then the contact determinacy of f is at most 2τ (f ) − mt(f ) + 2. Proof. cf. [BGM12, Cor. 1] Theorem 1.1.12 (Boubakri, Greuel and Markwig). Let 0 6= f ∈ m 1. If f is right k-determined, then mk+1 ⊂ m · j(f ). In particular, f is an isolated singularity. 2. If f is contact k-determined, then mk+1 ⊂ hf i + m · j(f ). In particular, Rf is an isolated hypersurface singularity. Proof. cf. [BGM12, Thm. 4]

1.1.3

Newton nondegeneracy

In this paragraph we recall some notions of non-degeneracy which were introduced in [Kou76], [Wal99], [BGM12]. P Definition 1.1.13. Let f = α cα xα ∈ K[[x]] be a power series. 1. We denote by supp(f ) = {α|cα 6= 0} the support and the convex hull of the set [ (α + Rn≥0 ) α∈supp(f )

the Newton polyhedron Γ+ (f ) of f . We call the union Γ(f ) of its compact faces the Newton diagram of f . By Γ− (f ) we denote the union of all line segments joining the origin to a point on Γ(f ). We always assume that f ∈ m if not explicitly stated otherwise. 2. f is called convenient, if its Newton diagram of meets all coordinate axes. 3. A compact rational polytope P of dimension n − 1 in the positive orthant Rn≥0 is called a C-polytope if the region above P is convex and if every ray in the positive orthant emanating from the origin meets P in exactly one point. The Newton diagram of f is a C-polytope iff it is convenient.

1.1 Hypersurface singularities

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Following Kouchnirenko, Wall, Boubakri, Greuel, Markwig we make the following definitions. P Definition 1.1.14. Let f = α cα xα ∈ m be a power series, let P be a C-polytope and let ∆ be a face of P . P 1. We denote by f∆ := in∆ (f ) := α∈∆ cα xα the initial form or principal part of f along ∆. We call f non-degenerate ND along ∆ if the Jacobian ideal j(in∆ (f )) has no zero in the torus (K ∗ )n . f is then said to be Newton non-degenerate NND if f is non-degenerate along each face (of any dimension) of the Newton diagram Γ(f ) (unlike Wall we do not require f to be convenient). 2. A face ∆ is called an inner face of P if it is not contained in any coordinate hyperplane. T 3. For each point q = (q1 , . . . , qn ) ∈ K n , set Hq = qi =0 {xi = 0} ⊂ Rn . We call f inner non-degenerate IND along ∆ if for each zero q of the Jacobian ideal j(in∆ (f )) the polytope ∆ contains no point on Hq . f is called inner Newton non-degenerate INND w.r.t. a C-polytope P if no point of supp(f ) lies below P and f is IND along each inner face of P . We call f simply inner Newton non-degenerate INND if it is INND w.r.t some C-polytope. 4. We call f weakly non-degenerate WND along ∆ if the Tjurina ideal tj(in∆ (f )) has no zero in the torus (K ∗ )n , and f is called weakly Newton non-degenerate WNND if f is weakly non-degenerate along each top-dimensional face of Γ(f ). Note that NND implies WNND. The following remark [BGM12, Remark 3] and [GN11] gives facts on and relations between the different types of non-degeneracy. Remark 1.1.15. For any occuring C-polytope P and power series f we assume that no point in supp(f ) lies below P . 1. Each of the non-degeneracy conditions introduced above only depends the principal part inP (f ) of f w.r.t. P . 2. Obviously ND along ∆ implies WND along ∆ and both are equivalent in characteristic zero, or, more generally, if char(K) does not divide the weighted degree of in∆ (f ). 3. WND along ∆ is strictly weaker than ND along ∆ in positive characteristic, but they are equivalent in characteristic zero. Moreover, WNND imposes only conditions on the facets (top dimensional faces) of Γ(f ) while NND does so for all faces of any dimension. E.g. f = x3 + y 2 with char(K) = 3 is WNND but not NND, since f is not ND along ∆ = {(3, 0)}. 4. If f is IND along ∆, then f is ND along ∆, but the converse is not true in general. E.g. f = x2 y 2 + y 4 and ∆ the line segment from the points (4, 0) to (0, 4), then f satisfies ND along ∆, but not IND. 5. If ∆ does not meet any coordinate hyperplane, then f is ND along ∆ if and only if it is IND along ∆.

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Chapter 1. Preliminaries 6. By (4.) NND does not imply INND but also INND need not imply NND since it only imposes conditions on the inner faces, see (7.). However in the planar situation NND + nearly convenient (i.e. µN < ∞) imply INND ([GN11, Cor 2.18]). Moreover they are equivalent if char(K) = 0 ([GN11, Cor 2.17]). 7. f can be convenient and INND without satisfying NND. E.g. f = (x + y)2 + xz + z 2 ∈ K[[x, y, z]] with char(K) 6= 2, then Γ(f ) has a unique facet ∆ with f = in∆ (f ) and 0 is the only zero of j(f ). Thus f is INND since no other face is inner. But f is not ND along the line segment from (2, 0, 0) to (0, 2, 0) which is a face of Γ(f ), i.e. f is not NND (see also [Kou76]). 8. If f is NND and k · ei ∈ Γ(f ), where ei is the i-th standard basis vector of Rn , then char(K) does not divide k. 9. f satisfies IND at an inner vertex α = (α1 , . . . , αn ) of P if and only if α is a vertex of Γ(f ) and some αi is not divisible by char(K).

10. In characteristic zero each of the above non-degeneracy conditions is a generality condition in the sense that fixing a C-polytope P then, among all polynomials f with supp(f ) ⊂ P , there is a Zariski open dense subset which satisfies the nondegeneracy condition. In positive characteristic some additional assumptions on the C-polytope P are necessary, like that not all coordinates of a vertex should be divisible by the characteristic. For any compact polytope Q in Rn≥0 we denote by Vk (Q) the sum of the kdimensional Euclidean volumes of the intersections of Q with the k-dimensional coordinate subspaces of Rn and, following Kouchnirenko, we then call µN (Q) =

n X

(−1)n−k k!Vk (Q)

k=0

the Newton number of Q. For a power series f ∈ K[[x]] we define the Newton number of f to be m µN (f ) = sup{µN (Γ− (fm )) | fm := f + xm 1 + . . . + xn , m ≥ 1}.

Note that, if f is convenient then µN (f ) = µN (Γ− (f )). Using the semi-continuity of the Milnor number, Gwozdziewicz in [Gw08] showed that the Newton number is monotonically increasing. That is, if f, g are covenient and Γ− (f ) ⊂ Γ− (g) then µN (f ) ≤ µN (g). Notice that the converse is not true in general. In Chapter 2 we give a charactization of the equality µN (f ) = µN (g) in the planar situation. Kouchnirenko in [Kou76] introduced the notion of NND and showed that if f is NND and convenient then the Newton number is equal to the Milnor number. More precisely,

1.2 Plane curve singularities

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Theorem 1.1.16. For f ∈ K[[x]] we have µN (f ) ≤ µ(f ), and if f is NND and convenient then µN (f ) = µ(f ) < ∞. Proof. cf. [Kou76]. Kouchnirenko also proved that the condition “convenient” is not necessary in Theorem 1.1.16 if char(K) = 0. The authors in [BGM12] show that in the planar case Kouchnirenko’s result holds in arbitrary characteristic without the assumption that f is convenient (allowing µ(f ) = ∞): Proposition 1.1.17. Suppose that f ∈ K[[x, y]] is NND, then µN (f ) = µ(f ). Proof. cf. [BGM12, Prop. 4]. Since Theorem 1.1.16 does not cover all semi-quasihomogeneous singularities, Wall introduced the condition INND (denoted by NPND* in [Wal99]). Using Theorem 1.1.16, Wall proved the following theorem for K = C which was extended to the field K of arbitrary characteristic in [BGM12]. Theorem 1.1.18. If f ∈ K[[x]] is INND, then µ(f ) = µN (f ) < ∞. Proof. cf. [Wal99], [BGM12]. In general it is unknown whether the converse of the theorem is true or not. However we show in Chapter 2 that this is true at least in the planar case.

1.2 1.2.1

Plane curve singularities Parametrization

Definition 1.2.1. Let 0 6= f ∈ m ⊂ K[[x, y]]. Then Rf = K[[x, y]]/hf i (or f ) is called a plane curve singularity. There is a unique (up to multiplication with units) decomposition f = f1ρ1 · . . . · frρr , with fi ∈ m irreducible in K[[x, y]]. fiρi resp. Rfiρi is called the i-th branch of f resp. of Rf . From now on we assume that f is reduced, i.e. ρi = 1 for i = 1, . . . , r. Let R = Rf be a plane curve singularity and let R0 be its integral closure (in ∼ the Lr total quotient ring Quot(R)). It follows from [GLS06], [Cam80] that R0 = i=1 K[[t]], where r is the number of branches of R. Definition 1.2.2. Let 0 6= f ∈ m ⊂ K[[x, y]] be reduced and R ,→ R0 be its normalization. A composition of the natural Lrprojection K[[x, y]] R, the normalization ∼ R ,→ R0 and an isomorphism R0 = i=1 K[[t]], ψ : K[[x, y]] R ,→ R0 ∼ =

r M

K[[t]]

i=1

is called a (primitive) parametrization of f (or of R). More precisely,

14

Chapter 1. Preliminaries 1. If f is irreducible, then a parametrization of f is given by a map ψ : K[[x, y]] −→ K[[t]] x 7→ x(t) y 7→ y(t). 2. If f decomposes into several branches, then a parametrization of R is given by a set of parametrizations of the branches. More precisely,L if f = f1 · . . . · fr ∼ is a decomposition of f into irreducible factors, then R0 = ri=1 K[[t]] is the normalization of R and a parametrization ψ of R can be represented as a matrix of the form:   x1 (t) y1 (t)  ..  ψ =  ... .  xr (t) yr (t) where for i = 1, . . . , r, (xi (t), yi (t)) represents a parametrization of the i-th branch.

Remark 1.2.3. If ψ as above is a parametrization of f then, for each permutation ρ of {1, 2, . . . , r},   xρ(1) (t) yρ(1) (t)   .. .. ψρ =   . . xρ(r) (t) yρ(r) (t) is also a parametrization of f . A parametrization of a reduced plane curve singularity has the following properties: Proposition 1.2.4. Let 0 6= f ∈ m ⊂ K[[x, y]] be reduced and ψ : K[[x, y]] R ,→ Lr R0 ∼ = i=1 K[[t]] be its parametrization. Then (i) ker(ψ) = hf i. 0

(ii) L ψ satisfies the following universal factorization property: Each ψ : K[[x, y]] → 0 r i=1 K[[t]], ψ (f ) = 0, factorizes Lr in a unique Lrway through ψ, that is there exists the unique morphism φ : i=1 K[[t]] → i=1 K[[t]] making the following diagram commutative: K[[x, y]]

ψ

ψ0

0

/

Lr

K[[t]]

i=1

L&

r i=1

φ

K[[t]]

Moreover, if ψ is also a parametrization of f , then φ is an isomorphism. Proof.

(i) is obvious.


15

(ii) follows from the universal factorization property of the normalization which is induced by the universal factorization property of the normalization of schemes cf. [Har77, II, Ex. 3.8]. We introduce now the correspondence between irreducible plane curve singularities and primitive elements of (K[[t]])2 . A couple (x(t), y(t)) in (K[[t]])2 is called primitive if for any couple (x0 (t), y 0 (t)) and some φ : K[[t]] → K[[t]]: x(t) = x0 (φ(t)), y(t) = y 0 (φ(t)) =⇒ φ(t) is an isomorphism. An element

   ψ1 x1 (t) y1 (t)    ..  ψ =  ...  =  ... .  ψr xr (t) yr (t) 

is called primitive if all ψi are primitive and the for all i 6= j there is no φ(t) ∈ K[[t]] such that ψi (φ(t)) = ψj (t). Proposition 1.2.5. (i) A couple (x(t), y(t)) ∈ (K[[t]])2 is a parametrization of some irreducible plane curve singularity, if and only if it is primitive. (ii) An element ψ ∈ R02 is a parametrization of some reduced plane curve singularity, if and only if it is primitive. Proof. (i) Assume first that (x(t), y(t)) is primitive. We consider the algebraic homomorphism determined by ψ : K[[x, y]] −→ K[[t]] x 7→ x(t) y 7→ y(t). Then ker(ψ) is a prime ideal and it is actually principal, i.e. ker(ψ) = hf i for some irreducible f ∈ K[[x, y]]. Let (x0 (t), y 0 (t)) be a parametrization of f . By the universal property of a parametrization (Proposition 1.2.4), there exists a homomorphism φ : K[[t]] → K[[t]] such that x(t) = x0 (φ(t)) and y(t) = y 0 (φ(t)). Then φ(t) is an isomorphism since (x(t), y(t)) is primitive. Hence (x(t), y(t)) is also a parametrization of f . Now suppose that (x(t), y(t)) is a parametrization of an irreducible plane curve singularity f and that x(t) = x0 (φ(t)), y(t) = y 0 (φ(t)), for some couple (x0 (t), y 0 (t)) and some φ : K[[t]] → K[[t]]. Then Φ(f )(x0 (t), y 0 (t)) = f (x0 (φ(t)), y 0 (φ(t))) = f (x(t), y(t)) = 0.

16


It yields f (x0 (t), y 0 (t)) = 0. It then follows from the universal property (Proposition 1.2.4), there exists a homomorphism φ0 : K[[t]] → K[[t]] such that x0 (t) = x(φ0 (t)) and y 0 (t) = y(φ0 (t)). This implies x(t) = x(φ0 (φ(t))) and then ordx(t) = ordx(t) · ordφ0 (t) · ordφ(t). Hence, ordφ(t) = 1, i.e. φ is an isomorphism. This proves the claim. (ii) follows easily from (i) and Proposition 1.2.4. Proposition 1.2.6. Let char(K) = p ≥ 0 and let f ∈ K[[x, y]] be irreducible such that m := mt(f ) = ordf (0, y). Assume that m is not divisible by p, then f has a Puiseux parametrization, i.e. a parametrization of the form X (x(t)|y(t)) := (tm | ck tk ). k≥m

Moreover, there exists a unit u ∈ K[[x, y]] such that m Y f =u· (y − y(ξ j x1/m )), j=1

where ξ is a primitive m-th root of unity. Proof. cf. [Cam80, Thm. 3.5.1], [GLS06, Prop. I.3.4]

1.2.2

Parametrization equivalence

We define the notion of parametrization equivalence and show that it is equivalent to contact equivalence. We mention here the definitions of reparametrization and coordinate change, which are necessary to define parametrization equivalence. L Let R0 = ri=1 K[[t]]. Considering R0 as a K-algebra, then we call an automorphism φ ∈ AutK (R0 ) a reparametrization of R0 . By Lemma 1.2.7, after a permutation of components of R0 , φ is determiend by (φ1 , . . . , φr ) with φi ∈ Aut(K[[t]]), i.e. φi (t) = ui1 t + ui2 t2 + . . . ∈ AutK (K[[t]]), ui1 6= 0, i = 1, . . . , r. In the following we often assume that a parametrization φ is given by (φ1 , . . . , φr ) with φi ∈ Aut(K[[t]]). A coordinate change of K[[x, y]] is an automorphism of K[[x, y]]. That is, we can write a coordinate change Φ as follows Φ : K[[x, y]] −→ K[[x, y]] X x 7→ cij xi y j X y 7→ c0ij xi y j ,


17

c10 c01 ∈ K and det 6= 0. where c010 c001 Note that ψ is a parametrization of f if and only if ψ ◦ Φ is a parametrization of Φ(f ). Lr Lemma 1.2.7. Let R0 = i=1 K[[t]] and let φ ∈ AutK (R0 ). Then there exist a permutation σ of {1, . . . , r} and φ1 , . . . , φr ∈ AutK (K[[t]]) such that φ(ψ) = φ1 (ψσ(1) ), . . . , φr (ψσ(r) ) cij , c0ij

for ψ = (ψ1 , . . . , ψr ) ∈ R0 . Proof. Let ei be the i-th standard basis vector of K r , and let φ(ei t) =

r X

φij (t)ej .

j=1

Since φ is an automorphism, the determinant det(φij ) 6= 0. This implies that there exists a permutation σ 0 of {1, . . . , r} such that φiσ0 (i) 6= 0 for all i = 1, . . . , r. We first show that φ(ei t) = φiσ0 (i) eσ0 (i) , i.e. show that φik = 0 for k 6= σ 0 (i). In fact, since k 6= σ 0 (i), ei t · eσ0−1 (k) t = 0. Then 0 = φ(ei t · eσ0−1 (k) t) =

r X

φij · φσ0−1 (k)j ej .

j=1

It follows that φik · φσ0−1 (k)k = 0 and hence φik = 0 since φσ0−1 (k)k 6= 0. Since φ(ei t) = φiσ0 (i) eσ0 (i) one has r X φ(ψ) = φ( ψi ei ) i=1

=

r X

φiσ0 (i) (ψi )eσ0 (i)

i=1

=

r X

φj (ψσ(j) )ej

j=1

with σ := σ 0−1 and φj := φσ(j)j . Now we define the notion of parametrization equivalence. L Definition 1.2.8. 1. Let ψ, ψ 0 : K[[x, y]] → R0 = ri=1 K[[t]]. Then ψ is said to be equivalent to ψ 0 , ψ ∼ ψ 0 , if there exist a reparametrization φ ∈ AutK (R0 ) and a coordinate change Φ ∈ AutK (K[[x, y]]) such that the following diagram commutes: K[[x, y]] Φ

ψ

K[[x, y]]

/ R0

ψ0

φ

/ R0

2. Let f, g ∈ K[[x, y]] be reduced. Then f is said to be parametrization equivalent to g, f ∼p g, if there exist a parametrization ψ of f and a parametrization ψ 0 of g such that ψ ∼ ψ 0 .

18


Remark 1.2.9. 1. If f ∼p g, then for any parametrization ψ (resp. ψ 0 ) of f (resp. g) we have ψ ∼ ψ 0 by Proposition 1.2.4(ii). 2. Assume that     ψ1 (t) ψ10 (t)     ψ(t) =  ...  ∼ ψ 0 (t) =  ...  . ψr0 (t)

ψr (t)

It follows from Lemma 1.2.7 that there exists a permutation σ of {1, . . . , r} such 0 that ψi (t) ∼ ψσ(i) (t) for all i. Moreover Proposition 1.2.10. Let f, g be two given plane curve singularities. Then f ∼p g ⇔ f ∼c g. Proof. Firstly, suppose that f ∼p g. Then, f and g have the same L number of branches r, therefore they have the same normalization ring R0 = ri=1 K[[t]]. Let 0 ψ (resp. ψ ) be a parametrization of f (resp. g) such that ψ ∼ ψ 0 . Then there exist Φ ∈ AutK (K[[x, y]]) and φ ∈ AutK (R0 ) such that the following diagram commutes ψ

K[[x, y]] Φ

/

R0 φ

K[[x, y]]

ψ0

/ R0 0

By definition, ker(ψ) = hf i then φ ◦ ψ(f ) = φ(0) = 0. Hence, ψ ◦ Φ(f ) = φ ◦ ψ(f ) = 0 0, which implies that Φ(f ) ∈ ker(ψ ) = hgi. Thus, hΦ(f )i ⊂ hgi. Similarly, ψ ◦ Φ−1 (g) = φ−1 ◦ ψ 0 (g) = 0 since ker(ψ 0 ) = hgi. Then Φ−1 (g) ∈ ker(ψ) = hf i, that is hgi ⊂ hΦ(f )i. Hence hgi = hΦ(f )i and then f ∼c g. Conversely, suppose that f ∼c g, then in particular, there exists an isomorphism of K-algebras ϕ : K[[x, y]]/hf i → K[[x, y]]/hgi. By the Lifting Lemma (cf. [GLS06, Lemma 1.23]), there exists an isomorphism Φ ∈ AutK (K[[x, y]]) such that the following diagram commutes: K[[x, y]] Φ

p1

/

K[[x, y]]/hf i ϕ

K[[x, y]]

p2

/

K[[x, y]]/hgi

Now, we consider the normalization maps n1 : K[[x, y]]/hf i ,→ R0 and n2 : Lr 0 K[[x, y]]/hgi ,→ R0 , where R0 = i=1 K[[t]]. Then ψ := n1 ◦ p1 (resp. ψ := n2 ◦ p2 ) is a parametrization of f (resp. g). Furthermore 0

ψ ◦ Φ(f ) = n2 ◦ p2 ◦ Φ(f ) = n2 ◦ ϕ ◦ p1 (f ) = n2 ◦ ϕ(0) = 0.


19

Then by the universal property of ψ, there exists an isomorphism φ : R0 → R0 making the following diagram commutative: ψ : K[[x, y]] Φ 0

p1

n1

K[[x, y]]/hf i

/

R0

ϕ

ψ : K[[x, y]]

//

p2

//

φ

K[[x, y]]/hgi n2 /

R0

The last commutative diagram means that ψ ∼ ψ 0 .

1.2.3

Intersection multiplicity and invariants

Definition 1.2.11. Let f ∈ K[[x, y]] be reduced and let ψ : K[[x, y]] R ,→ R0 ∼ = Lr K[[t]] be a parametrization of f . i=1 1. We call δ(f ) := dimK R0 /R the δ-invariant of f . 2. We introduce the valuation maps v := (v1 , ..., vr ) : R → (Z≥0 ∪ ∞)r , g 7→ ord(g(xi (t), yi (t)))i=1,...,r . Its image S(R) := S(f ) := v(R) is a semigroup, called the semigroup of values of f . 3. Let C := (R : R0 ) := {u ∈ R | uR0 ⊂ R} be the conductor ideal of R0 in R (cf. [ZS60]). Then C is an ideal of both R and R0 . So one has C = (tc1 )×· · ·×(tcr ) for some c := (c1 , . . . , cr ) ∈ Zr≥0 . We call c the conductor (exponent) of f . One obviously has c + Zr≥0 ⊂ S(f ) and c is the minimum element in S(f ) with this property w.r.t. the product ordering on Zr≥0 , i.e. the partial ordering given by: if α = (α1 , . . . , αr ), β = (β1 , . . . , βr ) ∈ Zr≥0 the α ≤ β if and only if αi ≤ βi for every i = 1, . . . , r. Definition 1.2.12. (1) Let g ∈ K[[x, y]] be irreducible and (x(t), y(t)) its parametrization. Then the intersection multiplicity of any f ∈ K[[x, y]] with g is given by i(f, g) := ordf (x(t), y(t)). If u is a unit then we define i(f, u) := 0. (2) The intersection multiplicity of f with a reducible power series g = g1 · . . . · gs , gi irreducible, is defined to be the sum i(f, g) := i(f, g1 ) + . . . + i(f, gs ). Proposition 1.2.13. Let f, g ∈ K[[x, y]]. Then i(f, g) = i(g, f ) = dim K[[x, y]]/hf, gi. Proof. cf. [GLS06, Prop. 3.12], the proof in [GLS06] was given for K = C but works in any characteristic.

20


Proposition 1.2.14. Let f, g ∈ K[[x, y]] be two reduced power series which have no factor in common. Then δ(f g) = δ(f ) + δ(g) + i(f, g). Proof. cf. [GLS06, Lemma. 3.32]. Proposition 1.2.15. Let f = f1 · . . . · fr with fi irreducible and let c its conductor. Then X X c(f ) = 2δ(f1 ) + i(f1 , fj ), . . . , 2δ(fr ) + i(fr , fj )) j6=1

=

c(f1 ) +

X

j6=r

i(f1 , fj ), . . . , c(fr ) +

j6=1

X

i(fr , fj ))

j6=r

and 2δ(f ) = c(f )1 + . . . + c(f )r . In particular if f is irreducible, then c(f ) = 2δ(f ). Proof. cf. [HeK71]. For more facts on the conductor see [GLS06], [Del87], [HeK71]. Definition 1.2.16 ([Cam80]). Let f ∈ K[[x, y]] be reduced and fi the irreducible components of f . Let γ ∈ K[[x, y]] be regular (i.e. mt(γ) = 1), we call β¯1 (f ) := sup{min i(fi , γ)|γ regular} i

the maximal contact multiplicity of f . Remark 1.2.17. Let f ∈ K[[x, y]] be irreducible. 1. If f is nonsingular, then β¯1 (f ) = ∞. 2. If f is singular, then β¯1 (f ) < ∞ and if f is irreducible then • β¯1 (f ) = β1 , where β1 is the first characteristic exponent of f (see, [Cam80], [BrK86]). • β¯1 (f ) = µ1 ·mt(f ), where the rational number µ1 is the first Newton coefficient for f (see, [Cam80], [BrK86]). Lemma 1.2.18 (Campillo). Let f ∈ K[[x, y]] be irreducible s.t. m := mt(f ) = ordf (0, y) and let n = ordf (x, 0). 1. If f is nonsingular (i.e. m = 1), then there exists a coordinate change Φ ∈ AutK K[[x, y]] such that Φ(f )(x, y) = y. 2. If f is singular, then there exists a coordinate change Φ ∈ AutK K[[x, y]] of the form Φ(x) = x + c1 y d1 + c2 y d2 + . . . , and Φ(y) = y, such that m - k, where k = ord Φ(f )(x, 0). In particular, if m | n then c1 6= 0 and d1 = n/m. If m - n then ci = 0 for all i ≥ 1. Moreover, β¯1 (f ) = k.


21

Proof. cf. [Cam80, Cor. 3.4.7]. The following proposition says that the δ-invariant, the conductor and the maximal contact multiplicity are invariant under contact equivalence, and by Proposition 1.2.10, they are also invariant under parametrization equivalence. Proposition 1.2.19. Let f, g ∈ K[[x, y]], let u, v ∈ K[[x, y]]∗ be unit and let Φ ∈ AutK (K[[x, y]]). Then i(f, g) = i(u · Φ(f ), v · Φ(g)). Moreover, if f ∼c g, then (i) δ(f ) = δ(g). (ii) c(f ) = c(g) (up to a permutation of the indices {1, . . . , r}). (iii) β¯1 (f ) = β¯1 (g). Proof. The equality i(f, g) = i(u · Φ(f ), v · Φ(g)) follows from Proposition 1.2.13 and the following isomorphism K[[x, y]]/hf, gi → K[[x, y]]/hu · Φ(f ), v · Φ(g)i, h 7→ Φ(h). (i) Since f ∼c g, as in the proof of Proposition 1.2.10, there exist isomor∼ = phisms Φ ∈ AutK (K[[x, y]]), φ ∈ AutK (R0 ) and ϕ : Rf = K[[x, y]]/hf i → Rg = K[[x, y]]/hgi such that the following diagram commutes ψ : K[[x, y]] Φ 0

p1

n1

Rf

/

R0

ϕ

ψ : K[[x, y]]

//

p2

//

Rg

φ

n2

/

R0

0

where ψ = n1 ◦ p1 is a parametrization of f , and ψ = n2 ◦ p2 is a parametrization of g. Then it can be easily checked that the map R0 /Rf → R0 /Rg ,

r 7→ φ(r)

is an isomorphism of K-algebras and hence δ(f ) = dimK R0 /Rf = dimK R0 /Rg = δ(g). (ii) Let f = f1 · . . . · fr be an irreducible decomposition of f . Since f ∼c g, there exist Φ ∈ AutK (K[[x, y]]) and u ∈ K[[x, y]]∗ such that g = u · Φ(f ). Then g = u · Φ(f1 · . . . · fr ) = u0 · Φ(f1 ) · . . . · u0 · Φ(fr ) = g1 · g2 · . . . · gr , where gi = u0 · Φ(fi ), and u0 is a unit s.t. u0r = u. It follows from parts i,ii that δ(fi ) = δ(gi ) and i(fi , fj ) = i(gi , gj ) for all i, j = 1, . . . , r and j 6= i. Hence by Proposition 1.2.15, X c(f )i = 2δ(fi ) + i(fi , fj ) j6=i

= 2δ(gi ) +

X j6=i

i(gi , gj ) = c(g)i .

22


(iii) Since f ∼c g, as in proof of (ii) there exist an irreducible decomposition f = f1 · . . . · fr of f , an irreducible decomposition g = g1 · . . . · gr of g, a unit u0 , an automorphism Φ ∈ AutK (K[[x, y]]) such that gi = u0 · Φ(fi ) for all i = 1, . . . , r. Now, for every regular plane curve γ, we have i(fi , γ) = i u0 · Φ(fi ), u0 · Φ(γ) = i gi , u0 · Φ(γ) . This implies that β¯1 (f ) = sup{min i(fi , γ)|γ regular} i = sup{min i gi , u0 · Φ(γ) |γ regular} i

≤ sup{min i(gi , γ 0 )|γ 0 regular}, i

since u0 · Φ(γ) is regular if γ is. Thus β¯1 (f ) ≤ β¯1 (g). Similarly we get β¯1 (f ) ≥ β¯1 (g) and hence β¯1 (f ) = β¯1 (g). For reduced plane curve f = f1 · . . . · fr with fi irreducible we define 1. mt(f ) := (mt(f1 ), . . . , mt(fr )) ∈ Zr the multi-multiplicity of f . 2. c(f ) := (c(f1 ), . . . , c(fr )) = (2δ(f1 ), . . . , 2δ(fr )) ∈ Zr the multi-conductor of f. These tuples are invariant under parametrization and contact equivalence as the following corollary shows. Corollary 1.2.20. If f ∼c g then mt(f ) = mt(g) and c(f ) = c(g) (up to a permutation of the indices {1, . . . , r}). Proof. Follows from Proposition 1.2.19. We recall that if f is a plane curve singularity then its Milnor number µ(f ) is dim K[[x, y]]/hfx , fy i, where fx , fy be the partials of f . Proposition 1.2.13 yields that the Milnor number can be computed as an intersection multiplicity: µ(f ) = i(fx , fy ). Moreover if K = C, the Milnor formula (see, [Mil68, Thm 10.5], or also [GLS06, Prop. 3.35]) gives a relation between the Milnor number, the δ-invariant: µ(f ) = 2δ(f ) − r(f ) + 1. Also, this is also true in characteristic zero . But in positive characteristic, it is in general not true as the following example shows: f = x3 + x4 + y 6 + y 7 ∈ K[[x, y]] with char(K) = 3. Then r(f ) = 1; µ(f ) = 18; δ(f ) = 6. In positive characteristic the equality holds under certain conditions of the Newton diagram, e.g. NND ([BGM12, Thm. 9]) or INND ([GN11, Cor. 3.16]). However without the assumption of Newton non-degeneracy one has at least an inequality as proven Deligne [Del73], see also [MHW01]: µ(f ) ≥ 2δ(f ) − r(f ) + 1. The difference of the two sides is measured by the so called Swan character which counts wild vanishing cycles that can only occur in positive characteristic. We will discuss more carefully about this topic in Chapter 2.


1.2.4 Let f =

23

Factorizations of plane curve singularities P

i,j

cij xi y j ∈ K[[x, y]] and Γ(f ) be its Newton diagram. We call fin := inΓ(f ) f =

X

cij xi y j

(i,j)∈Γ(f )

the initial part of f (w.r.t Γ(f )). Proposition 1.2.21. Let f ∈ m ⊂ K[[x, y]] be irreducible such that ordf (x, 0) = m and ordf (0, y) = n. Let (x(t), y(t)) be a parametrization of f . Then (a) ord(x(t)) = n and ord(y(t)) = m. (b) The Newton diagram of f is the straight line segment. (c) There exist ξ, λ ∈ K ∗ such that fin (x, y) = ξ · (xm/q − λy n/q )q , where q = gcd(m, n). Proof. cf. [Cam80, Lemma 3.4.3, 3.4.4, 3.4.5]. w = (n, m) of positive integers is called a weight. A polynomial f = P A pair i j c x y ∈ K[x, y] is called weighted homogeneous or quasihomogeneous of type i,j ij (w = (n, m); d) if d is a positive integer satisfying ni + mj = d, for each (i, j) ∈ supp(f ). Each formal power series f ∈ K[[x, y]] can decompose f into a sum w f = fdw + fd+1 + ...,

where fdw 6= 0 and flw is weighted homogeneous of type (w; d) for l ≥ d. We call fdw the initial part of f (w.r.t. w). If w = (1, 1) be a weight, then we call the initial part fdw the tangent cone of f and denoted by fd . We have then d = mt(f ). Note that for any weight w = (n, m) there exist the unique face ∆ of the Newton diagram Γ(f ) of f such that fdw = in∆ (f ). Conversely, for each edge E of the Newton diagram Γ(f ), there exist a weight w = (n, m) such that inE (f ) = fdw , where fdw is the initial part of f w.r.t. w. Then we call the rational number n/m the slope of ∆. Corollary 1.2.22. Let f ∈ m ⊂ K[[x, y]] be irreducible. Let w = (n, m) be a weight and let fdw the initial part of f w.r.t. w. Then fdw has only one irreducible factor which is one of the following: x; y; axm0 + by n0 with a, b 6= 0, n0 : m0 = n : m and gcd(n0 , m0 ) = 1.

24


Proposition 1.2.23. Let f ∈ K[[x, y]], let w = (n, m) be a weight with n, m coprime and let fdw the initial part of f w.r.t. w. Assume that fdw = g1r1 · . . . · gsrs with gj irreducible and hgj i pairwise distinct. Then there is a factorization of f : f = f¯1 · . . . · f¯s r such that gj j is initial part w.r.t. w of f¯j (which need not be irreducible).

Proof. Let f = f1 · . . . · fr with fi irreducible and let (fi )w di the initial part w.r.t. w of fi . Then w fdw = (f1 )w d1 · . . . · (fr )dr . For each j = 1, . . . , s we set Ij := {i = 1, . . . , r | (fi )w di ∈ hgj i} and note that if i ∈ Ij then by Corollary 1.2.22, r0

j 0 (fi )w di = ci · gj for some ci 6= 0, rj ≥ 1.

It is easy to verify that if f¯j :=

Y

fi /ci

i∈Ij

then f = f¯1 · . . . · f¯s is a factorization of f as required. Note that the tangent cone fd of f (with d = mt(f )) has always a factorization s Y fd = (ai x − bi y)ri i=1

with (bi : ai ) ∈ KP1 pairwise distinct. We call (bi : ai ) a tangent direction and (ai x − bi y) a tangent of f . The following lemma is just Proposition 1.2.25 with n = m = 1. Corollary 1.2.24. f can be factorized as f = f¯1 · . . . · f¯s such that (ai x − bi y)ri is the tangent cone of f¯j . We finish this section by giving a factorization of f associated its Newton diagram. Proposition 1.2.25. Let f ∈ K[[x, y]] and let Ei , i = 1, . . . , k be the edges of its Newton diagram. Then there is a factorization of f : f = monomial · f¯1 · . . . · f¯k such that f¯i is convenient, fEi = monomial × (f¯i )in . In particular, if f is convenient then f = f¯1 · . . . · f¯k .


25

Proof. We can prove for a convenient singularity f . Let f = f1 · . . . · fr be the irreducible decomposition of f . For each i, let ni /mi with gcd(ni , mi ) = 1 be the i slope of Ei , let wi = (ni , mi ) and let fdwii resp. (fj )w di,j the initial part fj w.r.t. wi for j = 1, . . . , r. We set i Ii := {j | (fj )w di,j is not monomial}

and f¯i :=

Y

fj .

j∈Ii

Then fEi =

fdwii

r Y Y i i ¯ = (fj )w = monomial × (fj )w di,j di,j = monomial × (fi )in j=1

j∈Ii

and we can see that {1, . . . , r} = tki=1 Ii . This proves the proposition.

1.2.5

Resolution of plane curve singularities

We collect several facts on resolution of (algebroid) plane curve singularities in [Cam80], [CGL07], [Hef03], [GLS06], [Zar65]. DefinitionL 1.2.26. 1. Let R be a graded ring, i.e. a ring together with a decomposition R = d≥0 R(d) into abelian groups such that R(d) · R(e) ⊂ R(d+e) . An element of R(d) is called homogeneous of degree d. An ideal I ⊂ R is called L homogeneous if it can be generated by homogeneous elements. Let R+ be the ideal d>0 R(d) . 2. We define the projective scheme Proj(R) to be the set of all homogeneous prime ideals p ⊂ R with R+ 6⊂ p. Definition 1.2.27. Let I ⊂ A be an ideal in a noetherian ring A. 1. We denote M M I d, I d /I d+1 and GrI (A) := grI (A) := d≥0

d≥0

where we set I 0 := A. 2. We call BlI (A) := Proj(GrI (A)) the blow-up of A at I and EI := Proj(grI (A)) its the exceptional divisor. 3. Let J be an ideal of A. We define a sheaf of ideals J 0 on BlI (A), called the strict transform of J as follows: On each open subset Spec(A[I/g]), J 0 := {a/g ν |a ∈ J and νI (a) ≥ ν}, where A[I/g] denotes the A-subalgebra of Ahgi , generated by a/g for all a ∈ I and νI (a) := max{ν|a ∈ I ν }. From now on, in this paragraph, P denotes K[[x, y]] and m its maximal ideal. R = K[[x, y]]/hf i denotes a plane curve singularity and mR its maximal ideal. Ri := K[[x, y]]/hfi i and mi its maximal ideal, where f = f1 · . . . · fr is an irreducible

26


factorization of f . Then grm (P ) = K[x, y], grmR (R) = K[x, y]/hfm i, and grmi (R) = K[x, y]/h(fi )mi i, i = 1, . . . , r, where Q (fi )mi resp. fm is the tangent cone of fi resp. f . Note that m = mt(f ) and fm = ri=1 (fi )mi . We consider the diagram of schemes E mi

/

_

Blmi (Ri )

/

E mR

/

_

BlmR (R)

/

E _ m

Blm (P )

induced by the following natural diagram of graded K-algebras grmi (Ri ) o o

grmR (R) o o OO

grm (P ) O

Grmi (Ri ) o o

GrmR (R) o o

Grm (P )

OO

O

Remark 1.2.28. 1. Since exceptional divisors are projectivizations of tangent cones, one element of Em corresponds to one point of KP1 , and the image of Emi in Em is one point Oi , counted mi times, where mi is the multiplicity of the branch Ri . An element of EmR is called an infinitely near point in the first infinitesimal neighbourhood of the origin m (which sometimes by geometrical reasons will be denoted by O) on f . Then an infinitely near point in the first infinitesimal neighbourhood of the origin O on f corresponds to a tangent direction (αi : βi ) ∈ KP1 of f , i.e. (fi )mi = ξ(βi x − αi y)mi , for some i = 1, . . . , s and some ξ ∈ K. 2. We may describe Blm (P ) as a union of two charts D+ (x) := Spec(P [y/x]) and D+ (y) := Spec(P [x/y]): Blm (P ) = D+ (x) ∪ D+ (y), P [y/x] denotes the P -subalgebra of Phxi , generated by y/x. 3. The projection π : Blm (P ) → SpecP induced by the ring homomorphism P ,→ Grm (P ), is called the blow-up map. One has then an isomorphism of schemes Em ∼ = π −1 (O), here again, O denotes the maximal ideal m. 4. Let O0 ∈ D+ (x) correspond to (α : β) ∈ Em . Then α 6= 0 and {x, xy − αβ } is a regular system of parameters for the local ring OBlm (P ),O0 . Thus setting u = ˆBlm (P ),O0 of OBlm (P ),O0 is isomorphic to K[[u, v]] and x, v = xy − αβ , the completion O the homomorphism of local rings T : P = K[[x, y]] −→ P 0 = K[[u, v]] induced by π is given by x 7→ u y 7→ u(v +

β ) α


27

5. Let J 0 be the strict transform of ideal hf i on Blm (P ) and let O0 ∈ D+ (x) correspond to (α : β) ∈ Em . Then the stalk J 0 O0 is generated by fÕ0 , which has a (local) equation at O0 in (chart 1) D+ (x): β fÕ0 (u, v) = u−m f (u, u(v + )), α where m = mt(f ). We call f˜ the strict transform of f . 6. For each infinitely near point Oi in the first infinitesimal neighbourhood on f , which corresponds to (αi : βi ), the normalization of L Ri0 = K[[u, v]]hf˜i i at Oi is R0i , and the normalization of R0 = K[[u, v]]hf˜i is R00 := si=1 R0i , where f˜i (resp. f˜) is the strict transform of fi (resp. of f ) at Oi . Further, notice that the parametrization P 0 −→ R0 of R0 is given by (ui (t), vi (t)), where ui (t) = xi (t), vi (t) :=

yi (t) βi − ∈ K[[t]], xi (t) αi

here, we assume that αi 6= 0. 7. Let f˜ the strict transform of f . Let Oi , i = 1, . . . , s be the infinitely near points in the first infinitesimal neighbourhood on f and let O0 ∈ Em . Then fÕ0 is unit iff O0 6∈ {O1 , . . . , Os }. 8. There exists a unique k ∈ N such that mt(f (i) ) > 1 (the notation mt(f (i) ) will be defined in definition 1.2.29) for all i < k and mt(f (k) ) = 1. Then the sequence f = f (0) → f (1) → · · · → f (k) , where f (i) is the strict transform of f (i−1) , is called the resolution of the plane curve singularity f . Definition 1.2.29. Let k ≥ 2, and assume that the infinitely near points O0 on f in the (k − 1)-th neighbourhood of O are defined. Assume also that the strict transform f (k−1) of f (k−2) is defined, where we set f := f (0) . We call each infinitely near point O00 on f (k−1) in the first infinitesimal neighbourhood of such a point O0 (k) an infinitely near point on f in the k-th neighbourhood of O. We call mO0 = mt(fO0 ) the multiplicity of O0 and denote mt(f (k) ) the maximum of mO0 , O0 running through all infinitely near point on f in the k-th neighbourhood of O. A point O0 is called an infinitely near points on f of the origin O if it is an infinitely near point on f in the k-th neighbourhood of O for some k ≥ 0. The following results were proven in [GLS06] for K = C but the proofs also work in any characteristic. Proposition 1.2.30. Let f˜ (resp. g˜) the strict transform of f (resp. g). Then X i(f, g) = mt(f ) · mt(g) + iO0 (fÕ0 , gÕ0 ). O0 ∈Em

In particular, i(f, g) = mt(f ) · mt(g) if f, g have no common tangent direction. Proof. cf. [GLS06, Proposition 3.21] Proposition 1.2.31. Let f ∈ m be reduced and f˜ its strict transform.

28


(i) δ(f ) =

X mt(f )(mt(f ) − 1) + δ(fÕ0 ). 2 O0 ∈E m

In particular, if f is irreducible and Q its the unique infinitely near points in the first neighbourhood of the origin, then δ(f ) =

mt(f )(mt(f ) − 1) + δ(f˜Q ). 2

(ii) The δ-invariant of f can be computed as δ(f ) =

X mq (mq − 1) q

2

,

where q runs through all infinitely near points on f of the origin O, and mq its multiplicity. Proof. cf. [GLS06, Proposition 3.34]

Chapter 2 Plane Curve Singularities In this chapter we study invariants of plane curve singularities: Milnor number, delta-invariant, gamma-invariant and their relations. We show that every reduced plane curve singularity is parametrization finitely determined and moreover that the parametrization determinacy is less than the conductor plus 1.

2.1

Milnor number

In the following we consider only the case of plane curve singularities. The main result of this section says that for f ∈ K[[x, y]], the condition µ(f ) = µN (f ) < ∞ is equivalent to f being INND (Theorem 2.1.10). In characteristic zero this is also equivalent to f being NND and µN (f ) < ∞ (Corollary 2.1.14). However, in positive characteristic, this is in general not true as the following example shows: Example 2.1.1. f = x3 + xy + y 3 in characteristic 3 satisfies µ(f ) = µN (f ) = 1 but f is not NND. Remark 2.1.2. Let f ∈ K[[x, y]] be convenient and Ai = (ci , ei ), i = 0, . . . , k the vertices of Γ(f ) with c0 = ek = 0, ci < ci+1 and ei > ei+1 . Then µN (f ) = 2V2 (Γ− (f )) − ck − e0 + 1. Lemma 2.1.3. Let f, g ∈ K[[x, y]] be convenient such that Γ− (f ) ⊂ Γ− (g). Then (a) µN (f ) ≤ µN (g). (b) The equality holds if and only if Γ− (f ) ∩ R2≥1 = Γ− (g) ∩ R2≥1 , where R2≥1 = {(x, y) ∈ R2 |x ≥ 1, y ≥ 1}. Note that Part (a) of the lemma holds true in many variables by [Biv09, Cor. 5.6]. Let us denote by Γ1 (f ) the cone joining the origin with Γ(f ) ∩ R2≥1 . (cf. Fig. 1). 29

30

Chapter 2. Plane Curve Singularities

e0 t J Jt J Jt Γ1 (f ) PP @ Γ(f ) P P @ q P @t H9 t (H 1 ((((H HHt ( ( t

0 1

ck

Fig. 1. Proof. First, we prove that µN (f ) = V2 (Γ1 (f )) + 1. It is easy to see that Γ1 (f ) divides Γ− (f ) into three parts whose volumes are ck /2, V2 (Γ1 (f )) and e0 /2. Therefore µN (f ) = 2V2 (Γ− (f )) − ck − e0 + 1 = 2V2 (Γ1 (f )) + 1. (a) Clearly, if Γ− (f ) ⊂ Γ− (g) then Γ1 (f ) ⊂ Γ1 (g) and hence µN (f ) = V2 (Γ1 (f )) + 1 ≤ 2V2 (Γ1 (g)) + 1 = µN (g). (b) follows easily from the formula µN (f ) = 2V2 (Γ1 (f )) + 1. We recall that for fixed a weight w = (n, m), each f ∈ K[[x, y]] can be decomposed into a sum w f = fdw + fd+1 + ..., where fdw 6= 0 and each flw , l ≥ d is weighted homogeneous of type (w; l) (see Chapter 1). fdw is called the initial part of f w.r.t. w. For each series ϕ(t) = c1 tα1 + c2 tα2 + . . . with c1 6= 0, α1 < α2 < . . ., we denote LT(ϕ(t)) := c1 tα1 and LC(ϕ(t)) := c1 . Lemma 2.1.4. Let w = (n, m) be a weight. Let x(t), y(t) ∈ K[[t]] with LT(x(t)) = w atα and LT(y(t)) = btβ such that α : β = n : m. Let fdw + fd+1 + . . ., be a w-weighted homogeneous decomposition of f ∈ K[[x, y]]. Then ordf (x(t), y(t)) ≥ dα/n. Equality holds if and only if fd (a, b) 6= 0. Proof. We can write x(t) = tα (a + u(t)) and y(t) = tβ (b + v(t)), where ordu(t) > 0 and ordv(t) > 0. Then X flw (x(t), y(t)) = cij (tα (a + u(t)))i (tβ (b + v(t)))j ni+mj=l lα

= t n flw (a + u(t), b + v(t)). Thus ordflw (x(t), y(t)) ≥ lα/n and hence ordf (x(t), y(t)) ≥ dα/n. Since fdw (a + u(t), b + v(t)) = fdw (a, b) + th(t) for some power series h, ordf (x(t), y(t)) = ordfdw (x(t), y(t)) = dα/n iff fdw (a, b) 6= 0.

2.1 Milnor number

31

Lemma 2.1.5. Let f ∈ K[[x, y]] be convenient such that Γ(f ) has only one edge. Let m = ordf (x, 0), n = ordf (0, y) and f = f1 · . . . · fr a factorization of f into its branches (irreducible factors). (a) Let (xj (t), yj (t)) be a parametrization of fj , j = 1, . . . , r with LT(xj (t)) = aj tαj and LT(yj (t)) = bj tβj . Then fin (aj , bj ) = 0, αj : βj = n : m and α1 +· · ·+αr = n. (b) Let a, b ∈ K ∗ such that fin (a, b) = 0. Then there is a parametrization (x(t), y(t)) of a branch of f satisfying LC(x(t)) = a and LC(y(t)) = b. Proof. Let w = (n, m) and let f = fdw be the initial part of f w.r.t. w. Then fdw = fin since Γ(f ) has only one edge.Q (a) It is easily verified that fin = (fj )in and that (fj )in is also a w-weighted homogeneous polynomial of some order dj . By Proposition 1.2.21(a), ordfj (x, 0) = βj and ordfj (0, y) = αj , i.e. xβj and y αj are monomials of (fj )in . Thus nβj = dj = mαj and hence αj : βj = n : m. dα Since f (xj (t), yj (t)) = 0, i.e. ordf (xj (t), yj (t)) = +∞ > n j , Lemma 2.1.4 yields that fdw (aj , bj ) = 0, i.e. fin (aj , bj ) = 0. Now, by the definition of intersection multiplicity we have n = i(f, x) =

r X

ordxj (t) =

j=1

r X

αj .

j=1

(b) Since f = g · h implies fin = gin · hin , it suffices to prove part (b) for the irreducible case. Then by Proposition1.2.21(c), there exist ξ, λ ∈ K ∗ such that 0

0

fin (x, y) = ρ · (xm − λy n )q , where q = (m, n); m0 = m/q and n0 = n/q. It is impossible that the characteristic p divides both m0 and n0 since (m0 , n0 ) = 1. We may assume that p does not divide n0 . Let (¯ x(t), y¯(t)) be a parametrization of f . Combining Proposition 1.2.21(a) and Lemma 2.1.4 we obtain that LT(¯ x(t)) = a ¯tn and LT(¯ y (t)) = ¯btm for some a ¯, ¯b ∈ K ∗ with fin (¯ a, ¯b) = 0. Set p 0 0 0 a := {ξi |i = 1, . . . , n0 }. g(y) := am − λy n and n a/¯ Then

0 0 0 g(¯bξim ) = fin (a, ¯bξim ) = fin (¯ aξin , ¯bξim ) = ξid fin (¯ a, ¯b) = 0. 0

0

Since (m0 , n0 ) = 1, it is easy to see that ξim 6= ξjm for all i 6= j. Thus the set 0 {¯bξim |i = 1, . . . , n0 }

contains all of roots of g. Since 0 = fin (a, b) = ρ · g(b)q , g(b) = 0. Then there is an p m0 q ¯ index i0 such that b = bξi0 . Choose a in ξi0 and put x(t) = x¯(t) and y(t) = y¯(t), we get LC(x(t)) = a and LC(y(t)) = b.

32


P Definition 2.1.6. Let f = cij xi y j ∈ K[[x, y]] be such that (0, n) is the vertex on the y-axis of Γ(f ). Let (1, j1 ) be the intersection point of Γ(f ) and the line x = 1. We define f to be ND1 along (0, n) if either char(K) = p = 0 or if p 6= 0 then p 6 |n or j1 ∈ N and the coefficient c1j1 of xy j1 in f is different from zero. ND1 along (m, 0), with (m, 0) the vertex on the x-axis of Γ(f ), is defined analogously. f is called ND1 along inner faces if it is ND. f is called NND1 if it is convenient and ND1 along all faces of Γ(f ). P Proposition 2.1.7. Let f = cij xi y j ∈ K[[x, y]] be convenient and let (0, n) (resp. (m, 0)) be the vertex on the y-axis (resp. on the x-axis) of Γ(f ). Assume that f is not ND1 along the point (0, n) or (m, 0) then µ(f ) > µN (f ). Proof. We consider only (0, n) since (m, 0) is analogous. Let (1, j1 ) be the intersection point of Γ(f ) and the line x = 1. The assumption that f is not ND1 along the point (0, n) implies that p|n and c1j1 = 0. Putting g(x, y) = f (x, y) − c0n y n one then has µ(f ) = µ(g) and Γ− (f ) ⊂ Γ− (g). On the other hand, it is easy to see that (1, j1 ) ∈ Γ+ (f ) \ Γ+ (g). This means Γ− (f ) ∩ R2≥1 $ Γ− (g) ∩ R2≥1 . It hence follows from Lemma 2.1.2 that µN (g) > µN (f ). Thus µ(f ) = µ(g) ≥ µN (g) > µN (f ).

Proposition 2.1.8. Let f ∈ K[[x, y]] be convenient. If f is degenerate along some inner vertex of Γ(f ) then µ(f ) > µN (f ). Proof. Assume that f is degenerate along some vertex (i0 , j0 ) of Γ(f ) with i0 > 0 and j0 > 0. Then p 6= 0 and i0 and j0 are divisible by p. Put g(x, y) = f (x, y)−ci0 j0 xi0 y j0 , then j(f ) = j(g) and hence µ(f ) = µ(g). Clearly, Γ+ (g) does not contain the point (i0 , j0 ). Thus Γ− (f ) ∩ R2≥1 $ Γ− (g) ∩ R2≥1 . Lemma 2.1.2 hence implies that µN (g) > µN (f ). We then have µ(f ) = µ(g) ≥ µN (g) > µN (f ).

Proposition 2.1.9. Let f ∈ K[[x, y]] be convenient. If f is degenerate along some edge of Γ(f ) then µ(f ) > µN (f ). P Proof. Let f (x, y) = cαβ xα y β . Let fx , fy be the partials of f and put h(x, y) := xfx (x, y) + λyfy (x, y), where λ ∈ K is generic. Then h(x, y) =

X

(α + λβ)cαβ xα y β .

Thus supp(h) = supp(f ) \ (pN)2 and if p = 0 then supp(h) = supp(f ). Hence Γ+ (h) ⊂ Γ+ (f ). Case 1: f is ND along each vertex of Γ(f ).

2.1 Milnor number

33

Assume now that (i, j) is a vertex of Γ(f ). Since f is ND along (i, j), p = 0 or p 6= 0 and one of i, j is not divisible by p. Therefore (i, j) ∈ supp(f )\(pN)2 = supp(h) and then Γ+ (f ) ⊂ Γ+ (h). Hence Γ(h) = Γ(f ). Let Ei , i = 1, . . . , k be edges of Γ(h). By Proposition 1.2.25, we can write ¯1 . . . h ¯ k , where h ¯ i are convenient and hE (x, y)) = monomial × (h ¯ i )in . We h = h i denote by mi and ni the lengths of the projections of Ei on the horizontal and vertical axes. wi i Let wi = (ni , mi ) and let hw di be the initial part of h w.r.t wi . Then hdi = hEi . Since Ei is also an edge of Γ(f ), fdwii = fEi , where fdwii is the initial part of f w.r.t. wi and then X

i hw di =

(α + λβ)cαβ xα y β = x

ni α+mi β=di

∂f w ∂fdwi + λy di . ∂x ∂y 0

Let gdwi0 be the initialb part of yfy w.r.t. wi . It is easy to see that di ≥ di and i

0

di = di iff y

w

∂fd i i

∂y

6= 0.

Claim 1. Let Ai−1 , Ai be the vertices of the edge Ei and let V2 (OAi−1 Ai ) be the volume of triangle OAi−1 Ai . Then di = 2V2 (OAi−1 Ai ). Proof. Let (ci , ei ) be the coordinates of Ai , i = 0, . . . , k. Then mi = ci − ci−1 and ni = ei−1 − ei . (cf. Fig. 2). ei−1 r Ati−1

r QQ Q QtAi ei !! ! ! ! t ! r

0

ci−1

ci

Fig. 2. Considering the rectangle (0, 0); (ci , 0); (ci , ei−1 ); (0, ei−1 ) we have 2V2 (OAi−1 Ai ) = = = =

2ci ei−1 − ci ei − ci−1 ei−1 − mi ni (ci−1 + mi )ei−1 + ci (ei + ni ) − ci ei − ci−1 ei−1 − mi ni mi ei−1 + ci ni − mi ni = mi (ei + ni ) + ci ni − mi ni mi ei + ni ci = di

This proves Claim 1. ¯ i , yfy ) ≥ di , and if f is degenerate along Ei then i(h ¯ i , yfy ) > di . Claim 2. i(h ¯ i,j of Proof. Let (xj (t), yj (t)), j = 1, . . . , r be parametrizations of the branches h α β ¯ i . Then by Lemma 2.1.5, we have LT(xj (t)) = aj t j and LT(yj (t)) = bj t j with h ¯ i (aj , bj ) = 0, αj : βj = ni : mi for all j = 1, . . . , r and α1 + . . . + αr = ni . aj , bj ∈ K ∗ , h 0

d α

It follows from Lemma 2.1.4 that ord(yfy )(xj (t), yj (t)) ≥ ini j for all j = 1, . . . , r. Thus 0 r r X X di αj 0 ¯ i(hi , yfy ) = ord(yfy )(xj (t), yj (t)) ≥ = di ≥ di . ni j=1 j=1

34

Chapter 2. Plane Curve Singularities Assume that f is degenerate along Ei then there exist a, b 6= 0 such that ∂fdwi ∂fdwi x (a, b) = y (a, b) = 0. ∂x ∂y

Therefore hdi (a, b) = 0. Lemma 2.1.5 implies that there is a parametrization of a ¯ i such that LT(¯ branch of h x(t)) = atα and LT(¯ y (t)) = btβ . We may assume that ¯ i,1 . Then α = α1 and β = β1 . (¯ x(t), y¯(t)) is a parametrization of the branch h ¯ i , yfy ) > di , we may restrict to the case that di 0 = di , because of the To show i(h w ¯ i , yfy ) ≥ di 0 ≥ di . As di 0 = di then g w 0 (a, b) = y ∂fdi (a, b) = 0. Lemma inequality i(h ∂y

di

2.1.4 yields ord(yfy )(¯ x(t), y¯(t)) >

di α1 . ni

Thus ¯ i , yfy ) = ord(yfy )(¯ i(h x(t), y¯(t)) +

r X

ord(yfy )(xj (t), yj (t))

j=2

>

di α 1 + ni

r X j=2

0

di αj = di . ni

This proves Claim 2. It now follows from Claim 1 and Claim 2 that i(h, yfy ) ≥

k X

2V2 (OAi−1 Ai ) = 2V2 (Γ− (f )).

i=1

Hence µ(f ) = i(fx , fy ) = i(h, yfy ) − i(x, fy ) − i(fx , y) − 1 ≥ 2V2 (Γ− (f )) − (e0 − 1) − (ck − 1) − 1 = µN (f ). Moreover, if f is degenerate along some edge of Γ(f ) then µ(f ) > µN (f ) by Claim 1 and 2. This proves of Case 1. Case 2: In the general case, by propositions 2.1.8, 2.1.9 we may assume that f is ND along each inner vertex and ND1 along the two vertice on the axes of Γ(f ). For m sufficiently large and p 6 |m, we put X f¯m (x, y) = cαβ xα y β + xm + y m . (α,β)6∈(pN)2

Then µ(f¯m ) = µ(fm ) = µ(f ) and µN (f¯m ) ≥ µN (fm ) = µN (f ), where the inequality follows from Lemma 2.1.3. Claim 3. f¯m is degenerate along some edge of Γ(f¯).

2.1 Milnor number

35

Proof. By the assumption f is degenerate along some edge E of Γ(f ). If E is also an edge of Γ(f¯m ) then j(inE (f¯m ) = j(inE (f )) and hence f¯m is degenerate along E. If E is not an edge of Γ(f¯m ), then E must meet the axes since f is ND along each inner vertex of Γ(f ). We may assume that (0, n) is a vertex of E. We will show that ](supp(f¯m ) ∩ E) ≥ 2. Let (1, j1 ) be the intersection point of E and the line x = 1. Since f is ND1 along (0, n), either (0, n) ∈ supp(f¯m ) ∩ E or (1, j1 ) ∈ supp(f¯m ) ∩ E, i.e. supp(f¯m ) ∩ E 6= ∅ On the other hand, it is easy to see that ](supp(f¯m ) ∩ E) 6= 1 since f is degenerate along the edge E. Hence ](supp(f¯m ) ∩ E) ≥ 2. Let us denote by E¯ the convex hull of the set supp(f¯m ) ∩ E. Then E¯ is an edge of Γ(f¯m ) and j(inE¯ (f¯m ) = j(inE (f )). Thus f¯m is degenerate along E¯ since f is degenerate along the edge E, which proves Claim 3. Now, by definition, f¯m is ND along each vertex of Γ(f¯m ). Since f¯m is degenerate along some edge of Γ(f¯m ), applying the first case to f¯m , we get µ(f¯m ) > µN (f¯m ). Hence µ(f ) = µ(f¯m ) > µN (f¯m ) ≥ µN (f ). This proves Proposition 2.1.9. Theorem 2.1.10. Let f ∈ m ⊂ K[[x, y]] and let fm = f + xm + y m . Then the following are equivalent (i) µ(f ) = µN (f ) < ∞. (ii) µ(f ) < ∞ and fm is NND1 for some large integer number m. (iii) f is INND. Proof. (i) ⇒ (ii) : Since µ(f ) = µN (f ) < ∞ we have by definition of µN (f ) µ(fm ) = µ(f ) = µN (f ) = µN (fm ) < ∞. Combining Propositions 2.1.7, 2.1.8 and 2.1.9 we get the claim. (ii) ⇒ (iii) : Assume that µ(f ) < ∞ and fm is NND1. Firstly, it is easy to see that there is an M ∈ N such that Γ(f ) ⊂ Γ(fM ). It suffices to show f is INND w.r.t. Γ(fm ) for all m > M . We argue by contradiction. Suppose that it is not true. Then f is not IND along some edge ∆ of Γ(fm ) which meets the axes, since fm is NND1. We may assume that ∆ meets the axes at (0, n). Let (k, l) be the second vertex of ∆. We consider two cases: • If l = 0, i.e. Γ(fm ) has only one edge ∆. Then ∆ is also a unique edge of Γ(f ) and in∆ (f ) = in∆ (fm ). Since f is not IND along ∆, there exists (a, b) ∈ K \ {(0, 0)} which is a zero point of j(in∆ (f )). Beside, since fm is ND along ∆, either a = 0 or b = 0. Assume that a = 0 and b 6= 0. We will show that fm is not ND1 along (0, n). ∂in∆ (fm ) ∂g Firstly, we write in∆ (fm ) = c0n y n + x · g(x, y), then = ny n−1 + x · . ∂y ∂y Thus ∂in∆ (fm ) (0, b) = ny n−1 = 0 ⇒ p 6= 0 and p|n. ∂y

36


We now write in∆ (fm ) = c0n y n + c1j xy j + x2 · h(x, y), then ∂h ∂in∆ (fm ) = c1j y j + 2x · h(x, y) + x2 · . ∂x ∂x ∂in∆ (fm ) (0, b) = 0, c1j = 0. Hence fm is not ND1 along (0, n), a contradiction. ∂x • Assume that l > 0. If ∆ is also an edge of Γ(f ) then in∆ (f ) = in∆ (fm ). Since f is not IND along ∆, there exists (a, b) ∈ K × K ∗ being a zero of j(in∆ (f )). Since fm is ND along ∆, a = 0. Analogously as above fm is not ND1 along (0, n) and we get a contradiction. Assume now that ∆ is not an edge of Γ(f ), i.e. m = n and x|f (x, y). Let P be the end point of Γ(f ) closest to y-axis. It follows from Γ(f ) ⊂ Γ(fM ) and m > M that P must be a vertex of ∆, i.e. P = (k, l). This implies f = xk · h(x, y). Since µ(f ) < ∞, k = 1. Then in∆ (f ) = c0n y n + c1l xy l and clearly f is always IND along ∆, a contradiction. Hence f is INND w.r.t. Γ(fm ) and then it is INND. (iii) ⇒ (i) : See Theorem 2.1.10. Since

Corollary 2.1.11. Let f ∈ K[[x, y]] and let M ∈ N such that Γ(f ) ⊂ Γ(fM ). Then f is INND if and only if it is INND w.r.t. Γ(fm ) for some (equivalently for all) m > M. Proof. One direction is obvious, it remains to show f is INND ⇒ f is INND w.r.t. Γ(fm ) for all m > M . We take m1 > M satisfying Theorem 2.1.10 and then f is INND ⇒ µ(f ) < ∞ and fm1 is NND1 ⇒ f is INND w.r.t. Γ(fm1 ). For each inner face ∆m of Γ(fm ), since m, m1 > M , there is an inner face ∆m1 of Γ(fm1 ) such that in∆m (f ) = in∆m1 (f ). Thus f is IND along ∆m since it is IND along ∆m1 . Hence f is INND w.r.t. Γ(fm ). Corollary 2.1.12. Let M ∈ N be such that Γ(f ) ⊂ Γ(fM ). Then Theorem 2.1.10 holds for each m > M . Remark 2.1.13. Let µ(f ) < ∞. Then M can be chosen as the maximum of n1 and m1 , where n1 = n if Γ(f ) ∩ {x = 0} = {(0, n)} and n1 = 2i1 if Γ(f ) ∩ {x = 0} = ∅ and Γ(f ) ∩ {x = 1} = {(1, i1 )}. Similarly we define m1 with x replaced by y. This remark and the previous corollaries are important for concrete computation. Proof of Corollary 2.1.12. Clearly, the equivalence (i) ⇔ (iii) does not depend on m and as in the proof of Theorem 2.1.10 the implication (ii) ⇒ (iii) holds for all m > M . It remains to show that f is INND ⇒ fm is NND1. By Corollary 2.1.11, it suffices to show that f is INND w.r.t. Γ(fm ) ⇒ fm is NND1. By contradiction, suppose that f is INND w.r.t. Γ(fm ) and fm is not NND1. Then f is not ND1 along some vertex of Γ(fm ) in the axes. Assume that f is not ND1 along (0, n) ∈ Γ(fm ). Then p 6= 0, p|n and Γ(fm ) ∩ {x = 1} ∩ supp(fm ) = ∅, i.e. (fm )in = c0n y n + x2 · h(x, y). This implies µ((fm )in ) = ∞. By Theorem 1.1.18, (fm )in is not INND and then fm is also not INND, a contradiction.

2.2 Delta-invariant

37

Corollary 2.1.14. Let K is a field of characteristic zero and f ∈ m ⊂ K[[x, y]]. Then the following are equivalent (i) µ(f ) = µN (f ) < ∞. (ii) f is INND. (iii) f is NND and µN (f ) < ∞. In particular, if f is convenient then (i)-(iii) are equivalent to (iv) f is NND. Proof. The implications (i) ⇒ (ii) and (iii) ⇒ (i) follow from Theorem 2.1.10 and Proposition 1.1.17. It remains to prove (ii) ⇒ (iii). Assume that f is INND. Then by Theorem 2.1.10, µN (f ) < ∞. We will show that f is ND along each vertex and each edge of Γ(f ). Since char(K) = 0, f is ND along each vertex of Γ(f ). Let ∆ be an edge of Γ(f ). Clearly, it is an inner edge of Γ(fm ), where m sufficiently large. Since f is INND, by Corollary 2.1.11 f is INND w.r.t. Γ(fm ). Then f is IND along ∆, and hence it is also ND along ∆. This implies f is NND. Corollary 2.1.15. If f is NND and µN (f ) < ∞ then f is INND. Proof. This follows from Proposition 1.1.17 and Theorem 2.1.10. Note that char(K) = 0 is only used to assure that f is ND along each vertex of Γ(f ) ∩ ({0} × N ∪ N × {0}). Hence, the last corollary holds also if p > 0 and p 6 |n if (0, n) = Γ(f ) ∩ {0} × N and p 6 |m if (m, 0) = Γ(f ) ∩ N × {0}. Example 2.1.1 shows that this condition is necessary. Conjecture 1. Let f ∈ K[[x]] = K[[x1 , . . . , xn ]]. Then the following are equivalent (i) µ(f ) = µN (f ) < ∞. (ii) f is INND.

2.2

Delta-invariant

We consider now another important invariant of plane curve singularities, the invariant δ and its combinatorial counterpart, the Newton invariant δN . We show that both coincide iff f is weighted homogeneous Newton non-degenerate (WHNND), a new non-degenerate condition introduced below. Let f ∈ m ⊂ K[[x, y]] be a power series. We recall that the multiplicity of f , denoted by mt(f ), to be the minimal degree of the homogeneous part of f . So f=

X k≥m:=mt(f )

fk (x, y),

38


where fk is homogeneous of degree k and fm 6= 0. Then fm decomposes into linear factors, s Y fm = (αi x − βi y)ri , i=1 1

with (βi : αi ) ∈ P pairwise distinct. We call fm the tangent cone and the points (βi : αi ), i = 1, . . . , s, the tangent directions of f . We fix a minimal resolution of the singularity computed via successively blowing up points, denote by Q → 0 that Q is an infinitely near point of the origin on f . If Q is an infinitely near point in the n-th neighbourhood of 0, we denote by mQ the multiplicity of the n-th strict transform of f at Q. If P is an infinitely near point in the l-th neighbourhood of 0, we denote by Q → P that Q is also an infinitely near point of P on the l-th strict transform f˜l of f at P . Note that if Q → P then n ≥ l and we set n(f˜l , Q) := n − l. In particular, we have n(f, Q) = n. Let E1 , . . . , Ek be the edges of the Newton diagram of f . We denote by l(Ei ) the lattice length of Ei , i.e. the number of lattice points on Ei minus one and by s(fEi ) the number of non-monomial irreducible (reduced) factors of fEi . We set P m (m −1) (a) ν(f ) := Q special Q 2Q , where an infinitely near point Q is special if it is the origin or the origin of the corresponding chart of the blowing up. (b) If f is convenient, we define Pk l(Ei ) V1 (Γ− (f )) + i=1 , δN (f ) := V2 (Γ− (f )) − 2 2 and otherwise we set δN (f ) := sup{δN (f (m) )|f (m) := f + xm + y m , m ∈ N} and call it the Newton δ-invariant of f . Pk (c) rN (f ) := i=1 l(Ei ) + max{j|xj divides f } + max{l|y l divides f }. P (d) sN (f ) := ki=1 s(fEi ) + max{j|xj divides f } + max{l|y l divides f }. Note that all these number depend (only) on the Newton diagram of f and hence are coordinate-dependent (for ν(f ) see Proposition 2.2.9). Proposition 2.2.1. For 0 6= f ∈ hx, yi we have r(f ) ≤ rN (f ), and if f is WNND then r(f ) = rN (f ). Proof. cf. [BGM12, Lemma 4] Let E be an edge of the Newton diagram of f . Then we can write fE as follows, s Y fE = monomial × (ai xm0 − bi y n0 )ri , i=1

where ai , bi ∈ K ∗ , (ai : bi ) pairwise distinct; m0 , n0 , ri ∈ N>0 , gcd(m0 , n0 ) = 1. It easy to see that s X s = s(fE ) and l(E) = ri . i=1

This implies s(fE ) ≤ l(E) and hence sN (f ) ≤ rN (f ). w Let w = (n0 , m0 ) and let f = fdw + fd+1 + . . . with fdw 6= 0 be the w-weighted homogeneous decomposition of f .

2.2 Delta-invariant

39

Definition 2.2.2. We say that f is weighted homogeneous non-degenerate (WHND) w along E if either ri = 1 for all i = 1, . . . , s or (ai xm0 − bi y n0 ) does not divide fd+1 for each ri > 1. f is called weighted homogeneous Newton non-degenerate (WHNND) if its Newton diagram has no edge or if it is WHND along each edge of its Newton diagram. Lemma 2.2.3. Let f ∈ K[[x, y]]. Then f is not WHNND if and only if there exist a, b ∈ K ∗ , w = (n, m) with gcd(m, n) = 1 such that fdw is divisible by (axm − by n )2 w w and fd+1 is divisible by (axm − by n ), where f = fdw + fd+1 + . . . is the w-weighted homogeneous decomposition of f . Proof. Straightforward from the above definition. Remark 2.2.4. (a) In [Lu87] the author introduced superisolated singularities to study the µ-constant stratum. We recall that f ∈ K[[x, y]] is superisolated if it becomes non-singular after only one blowing up. By ([Lu87, Lemma 1]), this is equivalent to: fm+1 (βi , αi ) 6= 0 for all tangent directions (βi : αi ) of f with ri > 1, where f = fm + fm+1 + . . . is the homogeneous decomposition of f and s Y fm = (αi x − βi y)ri . i=1

Note that this condition concerns all factors of fm including monomials. For WHNND singularities we require a similar condition, but for “all weights” and without any condition on the monomial factors of the initial term of the weigted homogeneous decomposition of f . (b) Since a plane curve singularity is superisolated iff it becomes non-singular after only one blowing up, we have δ(f ) = ν(f ) = m(m − 1)/2 and hence δ(f ) = δN (f ) = m(m − 1)/2, by Proposition 2.2.9. It follows from Theorem 2.2.12 that (c) A superisolated plane curve singularity is WHNND. (d) The plane curve singularity x2 + y 5 is WHNND but not superisolated. Proposition 2.2.5. With notations as above, f is WND along E if and only if s(fE ) = l(E) or, equivalently, iff ri = 1 for all i = 1, . . . , s. In particular, WNND implies WHNND. Proof. Firstly we can see that the equation s(fE ) = l(E) is equivalent to ri = 1 for all i = 1, . . . , s since s(fE ) = s and l(E) = r1 + . . . + rs . It remains to prove that f is WND along E iff ri = 1 for all i = 1, . . . , s. Assume that there is an i0 s.t. ∂fE ∂fE n0 0 , are divisible by (ai0 xm ri0 > 1. It is easy to see that fE , 0 − bi0 y0 ). Hence ∂x ∂y f is weakly degenerate (WD) along E. We now assume that f is weakly degenerate (WD) along E. Then there exist x0 , y0 ∈ K ∗ such that fE (x0 , y0 ) =

∂fE fE (x0 , y0 ) = (x0 , y0 ) = 0, ∂x ∂y

40


n0 0 and hence there exists an index i0 such that ai0 xm 0 − bi0 y0 = 0. We will show that ri0 > 1. In fact, if this is not true then fE (x, y) = (ai0 xm0 − bi0 y n0 ) · h(x, y) with h(x0 , y0 ) 6= 0. Since ∂fE fE (x0 , y0 ) = (x0 , y0 ) = 0, ∂x ∂y

this is impossible if p = 0 and implies that p divides m0 and n0 if p > 0. This contradicts the assumption gcd(m0 , n0 ) = 1. Let f ∈ K[[x, y]] and let Ei , i = 1, . . . , k be the edges of its Newton diagram. Then by Proposition 1.2.25 there is a factorization of f , f = monomial · f¯1 · . . . · f¯k , such that f¯i is convenient and fEi = monomial × (f¯i )in . Note that f¯i is in general not irreducible. On the other hand, f can be factorized into its irreducible factors as f = m1 · . . . · ml · f1 · . . . · fr , where mj are monomials, and fj are convenient. Proposition 2.2.6. (a) Let g, h ∈ K[[x, y]] such that f = g · h. If f is WHNND then so are both g and h. (b) With the above notations, the following are equivalent: (i) f is WHNND. (ii) f¯1 , . . . , f¯k are WHNND. (iii) f1 , . . . , fr are WHNND and (fi )in are pairwise coprime. Proof. (a) It suffices to show that if g is not WHNND then neither is f . In fact, since g is not WHNND, by Lemma 2.2.3, there exist a, b ∈ K ∗ and a weight w = (n, m) w with gcd(m, n) = 1 such that gcw is divisible by (axm − by n )2 and gc+1 is divisible by m n w w (ax −by ), where g = gc +gc+1 +. . . is the w-weighted homogeneous decomposition w w + . . . (resp. h = hw of g. Let f = fdw + fd+1 e + he+1 + . . .) be the w)-weighted homogeneous decomposition of f (resp. h). Then w w w w w fdw = gcw · hw e and fd+1 = gc · he+1 + gc+1 · he . w is divisible by (axm − by n ). This implies that fdw is divisible by (axm − by n )2 and fd+1 Again by Lemma 2.2.3, f is not WHNND. (b) It is easily verified that we may restrict to the case that f is convenient. The implication (i)⇒ (ii) follows from part (a). (ii)⇒ (iii): Assume that f¯1 , . . . , f¯k are WHNND. By part (a) we can deduce that f1 , . . . , fr are WHNND since for each i, fi is an irreducible factor of some f¯j . We now show that the (fi )in are pairwise coprime. By contradiction, suppose that (f1 )in and (f2 )in are not coprime. It follows from Proposition 1.2.25 that there exist a, b ∈ K ∗ , m, n ∈ N>0 with gcd(m, n) = 1 such that (axm − by n ) is the unique irreducible factor of (f1 )in and (f2 )in . Consequently, (f1 )in and (f2 )in are both wweighted homogeneous with w = (n, m). Assume that f1 resp. f2 is an irreducible factor of f¯j1 resp. f¯j2 for some j1 and j2 . Since (f¯j1 )in and (f¯j2 )in are weighted homogeneous, (f1 )in resp. (f2 )in is a factor of (f¯j1 )in resp. (f¯j2 )in . This implies

2.2 Delta-invariant

41

that (f¯j1 )in and (f¯j2 )in and therefore fEj1 and fEj2 are all w-weighted homogeneous. Then the edge Ej1 must coincide the edge Ej2 and hence f¯j1 = f¯j2 . It yields that the product g := f1 · f2 is a factor of f¯j1 . Now, we decompose g, f1 , f2 into their w-weighted homogeneous terms as follows: w w w w g = gcw + gc+1 + . . . , f1 = (f1 )w d1 + (f1 )d1 +1 + . . . , f2 = (f2 )d2 + (f2 )d2 +1 + . . . . w w w w w Then c = d1 + d2 , (f1 )w d1 = (f1 )in , (f2 )d2 = (f2 )in , gc = (f1 )d1 · (f2 )d2 and gc+1 = m n 2 w w w w (f1 )w d1 · (f2 )d2 +1 + (f1 )d1 +1 · (f2 )d2 . This implies that gc is divisible by (ax − by ) w is divisible by (axm − by n ). It follows from Lemma 2.2.3 that g is not and gc+1 WHNND and hence f¯j1 is also not WHNND by part (a) with g a factor of f¯j1 , which is a contradiction. (iii)⇒ (i): Suppose that f is not WHNND and that the (fi )in are pairwise coprime. We will show that fi is not WHNND for some i. Indeed, since f is not WHNND, by Lemma 2.2.3, there exist a, b ∈ K ∗ and a weight w = (n, m) w is divisible by with gcd(m, n) = 1 such that fdw is divisible by (axm − by n )2 and fd+1 m n w w (ax −by ), where f = fd +fd+1 +. . . is the w-weighted homogeneous decomposition w of f . Let (fi ) = (fi )w di + (fi )di +1 + . . . be the w-weighted homogeneous decomposition of fi , i = 1, . . . , r. Then we have

d=

r X

di ;

(fi )w di

= (fi )in ;

fdw

r Y = (fi )w di ;

i=1

w fd+1

i=1

r X Y w (fi )w = · (f ) l di +1 dl . i=1

l6=i

Q w Since fdw = ri=1 (fi )w di and since the (fi )di are pairwise coprime, there exists an i0 m n 2 w m n such that (fi0 )w di0 is divisible by (ax −by ) and (fl )dl is not divisible by (ax −by ) m n for all l 6= i0 . This implies that (fi0 )w di +1 is divisible by (ax − by ) since 0

w fd+1 = (fi0 )w di0 +1 ·

Y

(fl )w dl +

l6=i0

X

(fi )w di +1 ·

i6=i0

Y (fl )w dl . l6=i

Then fi0 is not WHNND by Lemma 2.2.3. Proposition 2.2.7. For 0 6= f ∈ hx, yi we have sN (f ) ≤ r(f ) and if f is WHNND then sN (f ) = r(f ). Proof. If f = xj y l · g(x, y) with g convenient, then sN (f ) = sN (g) + j + l and r(f ) = r(g) + j + l, so we may assume that f is convenient. Step 1. Assume first that Qr the Newton diagram Γ(f ) has only one edge E. Then we can see that fin = i=1 (fi )in . It follows from Proposition 1.2.25 that for each i, (fi )in has only one irreducible factor and therefore fin has at most r irreducible factors. This means that r ≥ sN (f ). If r(f ) > sN (f ), then there exist i 6= j such that (fi )in and (fj )in have the same factor. This means that (fi )in and (fj )in are not coprime. Then by Proposition 2.2.6, f is not WHNND. Step 2. Assume now that the Newton diagram Γ(f ) has k edges E1 , . . . , Ek . By

42


Proposition 1.2.25, f can be factorized as f = f¯1 · . . . · f¯k , where f¯j is convenient, its Newton diagram has only one edge and fEj = monomial·(f¯j )in for each j = 1, . . . , k. This implies that sN (f¯j ) = s(fEj ). Then we obtain r(f ) =

k X

r(f¯j ) ≥

j=1

k X

sN (f¯j ) =

j=1

k X

s(fEj ) = sN (f ).

j=1

Now we assume that r(f ) > sN (f ). Then there exists a j = 1, . . . , k such that ¯ r(fj ) > s(fEj ) = sN (f¯j ). It follows from Step 1 that f¯j is not WHNND. Hence f is not WHNND by Proposition 2.2.6, which proves the proposition. Proposition 2.2.8. For 0 6= f ∈ hx, yi we have sN (f ) ≤ r(f ) ≤ rN (f ), and both equalities hold if and only if f is WNND. Proof. The inequalities follow from Proposition 2.2.1 and Proposition 2.2.7. For each edge E of Γ(f ), by Proposition 2.2.5, f is WND along E iff s(fE ) = l(E). This implies that f is WNND if and only if sN (f ) = rN (f ) since s(fE ) ≤ l(E) and both sides are additive with respect to edges of Γ(f ). We investigate now the relations between ν(f ), δN (f ) and δ(f ), which were studied in [BeP00] and [BGM12]. Proposition 2.2.9. If f ∈ K[[x, y]] then δN (f ) = ν(f ). Proof. cf. [BGM12, Lemma 3]. Proposition 2.2.10. For 0 6= f ∈ hx, yi we have δN (f ) ≤ δ(f ), and if f is WNND then δN (f ) = δ(f ). Proof. cf. [BGM12, Prop. 5]. Hence WNND is sufficient but, by the following example, not necessary for δN (f ) = δ(f ). Example 2.2.11. Let f (x, y) = (x + y)2 + y 3 ∈ K[[x, y]]. Then f is not WNND but δN (f ) = δ(f ) = 1. This easy example shows also that WNND depends on the coordinates since x2 + y 3 is WNND. Note that f is WHNND. Now we prove that WHNND is necessary and sufficient for δN (f ) = δ(f ). Theorem 2.2.12. Let f ∈ K[[x, y]] be reduced. Then δ(f ) = δN (f ) if and only if f is WHNND. We will prove the theorem after three technical lemmas. Let E be an edge of the Newton diagram of f . We write s Y fE = monomial × (ai xm0 − bi y n0 )ri , i=1

where ai , bi ∈ K ∗ , (ai : bi ) pairwise distinct; m0 , n0 , ri ∈ N>0 , (m0 , n0 ) = 1.

2.2 Delta-invariant

43

Lemma 2.2.13. With the above notations, there exist an integer n and an infinitely near point Pn in the n-th neighbourhood of 0, such that (fñ )En (u, v) = monomial ×

s Y (ai u − bi v)ri , i=1

where fñ is a local equation of the strict transform of f˜ at Pn and En is some edge of its Newton diagram Γ(fñ ). Moreover, f is WHND along E if and only if fñ is WHND along En . Proof. We prove the lemma by induction on m0 + n0 . If m0 + n0 = 2, i.e. m0 = n0 = 1, then the claim is trivial. Suppose m0 + n0 > 2. Now we show the induction step. Since m0 + n0 > 2 and gcd(m0 , n0 ) = 1, m0 6= n0 . We may then assume that m0 < n0 . Then P1 := (1, 0) is a special infinitely near point of 0 and the local equation of f˜1 at P1 in chart 2, is: f (x1 y1 , y1 ) , where m = mt(f ). f˜1 (x1 , y1 ) = y1m Let w0 = (n0 , m0 ) and let f = fdw00 + fdw00+1 + . . . with fdw00 6= 0 be the w0 -weighted 1 homogeneous decomposition of f . It is easy to see that if we decompose f˜1 = (f˜1 )w e0 + 1 (f˜1 )w e0 +1 + . . . into the w1 -weighted homogeneous terms with w1 = (n0 − m0 , m0 ), then fdw00+ν (x1 y1 , y1 ) 1 , ∀ν ≥ 0. e0 = d0 − m · m0 and (f˜1 )w = e0 +ν y1m In particular, 1 (f˜1 )w e0 = monomial ×

s Y (ai xm0 − bi y n0 −m0 )ri . i=1

1 We denote by E1 the convex hull of the support of (f˜1 )w e0 . Clearly, E1 is an edge of Γ(f˜1 ). Since

1 (f˜1 )E1 = (f˜1 )w e0 = monomial ×

s Y (ai xm0 − bi y n0 −m0 )ri i=1

w0

fd +1 (x1 y1 ,y1 ) 1 0 , it follows that f is WHND along E iff f˜1 is also and since (f˜1 )w e0 +1 = y1m WHND along E1 . Hence the induction step is proven by applying the induction hypothesis to f˜1 .

The above lemma yields that QE,i := (bi : ai ), i = 1, . . . , s, are determined by fE and they correspond to tangent directions of fñ . Then they are infinitely near points in the first neighbourhood of Pn . Consequently, they are infinitely near points in the (n + 1)-th neighbourhood of 0. To compute the multiplicity mQE,i , we consider

44


the local equation of the strict transform fñ+1 of fñ at QE,i = (bi : ai ) in chart 2: fñ+1 (u1 , v1 ) = =

fñ ((u1 +

bi )v , v ) ai 1 1 e0 v1 w ˜ (fn )e0 ((u1 + abii )v1 , v1 ) v1e0

+

(fñ )w e0 +1 ((u1 +

bi )v , v ) ai 1 1

+ ... v1e0 bi bi = (fñ )w , 1) + v1 · (fñ )w , 1) + . . . , e0 (u1 + e0 +1 (u1 + ai ai ˜ w ˜ w where fñ = (fñ )w e0 + (fn )e0 +1 + . . . with (fn )e0 6= 0, is the (w-weighted) homogeneous decomposition of f (i.e. w = (1, 1)). Since ri (fñ )En (u, v) = (fñ )w e0 (u, v) = (ai u − bi v) · g(u, v) with g(bi , ai ) 6= 0,

we get bi bi fñ+1 (u1 , v1 ) = (ai u1 )ri · g(u1 + , 1) + v1 · (fñ )w , 1) + . . . , e0 +1 (u1 + ai ai with g(u1 + abii , 1) a unit. In the following, this equality will be used to compare the multiplicity mt(fñ+1 ) with 1. Lemma 2.2.14. With the above notations, (a) if f is WHND along E, then mQE,i = 1 for all i; (b) if f is not WHND along E, then mQE,i > 1 for some i. Proof. Note that in this proof we consider the weight w = (1, 1). (a) Since f is WHND along E, it follows from Lemma 2.2.13 that fn is WHND along En , i.e. either ri = 1 for all i or (ai u − bi v) is not a factor of (fñ )w e0 +1 for each ˜ ri > 1. If ri = 1, it is easy to see that mQE,i = mt(fn+1 (u1 , v1 )) = 1 for all i. If ˜ w ri > 1 and (ai u − bi v) is not a factor of (fñ )w e0 +1 . Then (fn )e0 +1 (bi , ai ) 6= 0. This bi implies that (fñ )w e0 +1 (u1 + ai , 1) is a unit. Hence mQE,i = mt(fñ+1 (u1 , v1 )) = 1. (b) Assume that f is not WHND along E. By Lemma 2.2.13, fñ is not WHND along En , i.e. there exists an i such that ri > 1 and (ai u − bi v) is a factor of (fñ )w e0 +1 . w ˜ Therefore (fn )e0 +1 (u, v) = (ai u − bi v) · h(u, v) and then (fñ )w e0 +1 (u1 +

bi bi , 1) = (ai u1 ) · h(u1 + , 1). ai ai

Hence mQE,i = mt(fñ+1 (u1 , v1 )) > 1. Lemma 2.2.15. With the above notations, if Q is not special, then there exists an edge E of Γ(f ) such that Q → QE,i for some i.

2.2 Delta-invariant

45

Proof. We will prove the lemma by induction on n(f, Q). First, since Q is not special, n(f, Q) ≥ 1. If n(f, Q) = 1, then Q is a tangent direction of f and we can write Q = (b : a), where (ax − by) is a factor of the tangent cone fm of f . Since Q is not special, fm is not monomial. This implies that there exists an edge E of Γ(f ) such that fE = fm . We can write s Y fE = fm = monomial × (ai x − bi y)ri i=1

with (b : a) = (b1 : a1 ), consequently Q = QE,1 . Now we prove the induction step. Suppose that n(f, Q) > 1. Then Q → P for some infinitely near point P in the first neighbourhood of 0. If P is not special, then as above, P = QE,1 for some edge E of Γ(f ) and hence Q → QE,1 . If P is special, we may assume that P = (0 : 1). Then the local equation of the strict transform f˜ of f at P in chart 2, is: f (uv, v) . f˜(u, v) = vm Since n(f˜, Q) = n(f, Q) − 1 and by induction hypothesis, there is an edge E 0 of Γ(f˜) such that s Y 0 0 ˜ fE 0 = monomial × (ai um0 − bi v n0 )ri , i=1

where ai , bi ∈ K ∗ , (ai : bi ) pairwise distinct; m00 , n00 , ri ∈ N>0 , gcd(m00 , n00 ) = 1 and w0 Q → QE 0 ,i for some i. Let us denote m0 = m00 , n0 = m00 +n00 and f = fdw0 +fd+1 +. . . be the w0 -weighted homogeneous decomposition of f with w0 = (n0 , m0 ). Then for each l > d, we have flw0 (uv, v) = vm

X

cαβ (uv)α v β−m

n0 α+m0 β=l

X

=

cαβ uα v α+β−m .

n00 α+m00 (α+β−m)=l−mm00 0

0

w w0 This implies that f˜ = f˜e 0 + f˜e+1 +. . . is the w00 -weighted homogeneous decomposition w00 fl (uv,v) of f˜ with w00 = (n00 , m00 ), where e = d − mm00 and f˜l−mm . Note that 0 = vm 0 0 w f˜e 0 = f˜E 0 . It is easy to see that

fdw0 (x, y)

=

0 x y m f˜ew0 ( , y)

y

s Y = monomial × (ai xm0 − bi y n0 )ri . i=1

Since E 0 is an edge of Γ(f˜E 0 ), f˜E 0 and then fdw0 (x, y) are not monomials. By E we denote the convex hull of the support of fdw0 . Then E is an edge of Γ(f ) and fE = fdw0 . Therefore QE,i = QE 0 ,i and hence Q → QE,i . Proof of Theorem 2.2.12. (=⇒): Assume f is not WHNND, then f is not WHND along some edge E of Γ(f ). By Lemma 2.2.13, there is an infinitely near point QE,i

46


of 0, such that mQE,i > 1. Clearly, QE,i is not special. This implies that δ(f ) > ν(f ) and hence δ(f ) > ν(f ) = δN (f ) due to Proposition 2.2.9. (⇐=): Assume now that f is WHNND. To show δ(f ) = δN (f ), it suffices to show that there is no infinitely near point Q of 0 such that Q is not special and mQ > 1. We argue by contradiction. Suppose that there is such an infinitely near point Q. By Lemma 2.2.15, there is an edge E of Γ(f ) such that Q → QE,i for some i, and then mQ ≤ mQE,i . Since f is WHND along E, it follows from Lemma 2.2.14 that mQE,i = 1. Hence mQ ≤ mQE,i = 1, which is a contradiction. If char(K) = 0 we have Milnor’s famous formula µ(f ) = 2δ(f ) − r(f ) + 1, where r(f ) is the number of branches of f . The formula is wrong in general if char(K) > 0 but still holds if f is NND by [BGM12, Thm. 9]. Using the general inequality µN (f ) = 2δN (f ) − rN (f ) + 1 ≤ 2δ(f ) − r(f ) + 1 ≤ µ(f ) from [BGM12], then Theorem 2.1.10, Proposition 2.2.1 and Proposition 2.2.10 imply Corollary 2.2.16. Let f ∈ K[[x, y]] be reduced. Then f is INND if and only if f is WNND and µ(f ) = 2δ(f ) − r(f ) + 1. Remark 2.2.17. (1) The difference wvc(f ) := µ(f ) − 2δ(f ) + r(f ) − 1 counts the number of wild vanishing cycles of (the Milnor fiber) of f (cf. [Del73], [MHW01], [BGM12]), which vanishes if char(K) = 0 or if f is INND. (2) wvc(f ) is computable for any given f . This follows since µ(f ) is computable by a standard basis computation w.r.t. a local ordering (cf. [GP08]) and δ(f ) and r(f ) are computable by computing a Hamburger-Noether expansion (cf. [Cam80]). Both algorithms are implemented in Singular (cf. [GPS05]). Example 2.2.18. Consider f = x(x − y)2 + y 7 and g = x(x − y)2 + y 7 + x6 and char (K) = 3. Using Singular we compute µ(f ) = 8, δ(f ) = 5, r(f ) = 3 and µ(g) = 8, δ(f ) = 4, r(g) = 2. We have wvc(f ) = 0, wvc(g) = 1, Γ(f ) = Γ(g) and f is not INND. This shows • INND is sufficient but not necessary for the absence of wild vanishing cycles, • the Newton diagram can not distinguish between singularities which have wild vanishing cycles and those which have not. Although we can compute the number of wild vanishing cycles, it seems hard to understand them. We like to pose the following Problem 1. Is there any “geometric” way to understand the wild vanishing cycles, distinguishing them from the ordinary vanishing cycles counted by 2δ−r+1? Is there at least a “reasonable” characterization of those singularities without wild vanishing cycles? Comment: see Theorem 2.4.4.

2.3 Gamma-invariant

2.3

47

Gamma-invariant

In this section, motiving by Problem 1, we introduce the gamma-invariant of plane curve singularities and its relations to the delta-invariants and the number of branches (Prop. 2.3.7). As a consequence we give a formula to compute the delta-invariant of an irreducible plane curve singularity f , in terms of intersection multiplicities in case the characteristic p is m-good for f (Cor. 2.3.8). Definition 2.3.1. Let f ∈ K[[x, y]] be reduced. The gamma-invariant of f , denoted by γ(f ), is defined as follows: (1) γ(x) := 0, γ(y) := 0 and γ(xy) := 1; (2) if f is convenient (see, Definition 1.1.13) then γ(f ) := i(f, αxfx + βyfy ) − i(f, x) − i(f, y) + 1, where (α : β) ∈ KP1 is generic; (3) if f = xk y l · g with 0 ≤ k, l ≤ 1 and g convenient, then γ(f ) := γ(xk y l ) + γ(g) + 2 · i(xk y l , g) − 1. By definition we can deduce that γ(u) = 1 and γ(u · f ) = γ(f ) for every unit u. But γ is not invariant under right equivalence. E.g. f = x3 + x4 + y 5 and g = (x + y)3 + (x + y)4 + y 5 in K[[x, y]] with char(K) = 3 and then f ∼r g, but γ(f ) = 8, γ(g) = 10. However, if the characteristic p is m-good (cf. Definition 2.3.3) for f then γ(f ) = γ(g) for all g is contact equivalent to f (see, Lemma 2.3.4). Lemma 2.3.2. Let f ∈ K[[x, y]] be reduced. Then (i) If f is irreducible and convenient, then γ(f ) = min{i(f, fx ) − i(f, y) + 1, i(f, fy ) − i(f, x) + 1}. (ii) If f = f1 · . . . · fr , then γ(f ) =

r X i=1

γ(fi ) +

X

i(fi , fj ) − r + 1.

j6=i

Proof. (i) Let (x(t), y(t)) be a parametrization of f . For (α : β) ∈ KP1 generic, we have γ(f ) = = = = =

i(f, αxfx + βyfy ) − i(f, x) − i(f, y) + 1 ord αx(t)fx (x(t), y(t)) + βy(t)fy (x(t), y(t)) − i(f, x) − i(f, y) + 1 min{ord x(t)fx (x(t), y(t)) , ord y(t)fy (x(t), y(t)) } − i(f, x) − i(f, y) + 1 min{i(f, xfx ) − i(f, x) − i(f, y) + 1, i(f, yfy ) − i(f, x) − i(f, y) + 1} min{i(f, fx ) − i(f, y) + 1, i(f, fy ) − i(f, x) + 1

48


(ii) Note that there are at most two of the fi being not convenient since f is reduced. Thus the proof falls naturally into 3 cases: Case 1: All the fi are convenient, then so is f and hence

γ(f ) = i(f, αxfx + βyfy ) − i(f, x) − i(f, y) + 1 r X = i(fi , αxfx + βyfy ) − i(fi , x) − i(fi , y) + 1 i=1

= =

r X i=1 r X

i(fi , αx

X ∂fi ∂fi + βy )+ i(fi , fj ) − i(fi , x) − i(fi , y) + 1 ∂x ∂y j6=i

γ(fi ) − 1 +

i=1

=

X

i(fi , fj ) + 1

j6=i

r X

γ(fi ) +

i=1

X

i(fi , fj ) − r + 1.

j6=i

Case 2: There is exactly one of the fi not convenient, say fr . We write fr = gr · xk y l with gr convenient. Since f1 , . . . , fr−1 and gr are convenient, due to the first case and by definition we obtain

γ(f ) = γ(f1 · . . . · fr−1 · gr · xk y l ) = γ(xk y l ) + γ(f1 · . . . · fr−1 · gr ) + 2 · i(xk y l , f1 · . . . · fr−1 · gr ) − 1 r−1 r−1 X X X k l i(fi , fj ) + γ(gr ) + 2 · i(gr , fi ) − r + 1 = γ(x y ) + γ(fi ) + i=1

i=1

j6=i

k l

k l

+2 · i(x y , f1 · . . . · fr−1 ) + 2 · i(x y , ·gr ) − 1 r−1 X X = γ(fi ) + i(fi , fj ) + γ(fr ) + 2 · i(fr , f1 · . . . · fr−1 ) − r + 1 =

i=1 r X i=1

j6=i

γ(fi ) +

X

i(fi , fj ) − r + 1.

j6=i

Case 3: There are exactly two of the fi not convenient, say fr−1 and fr . Since f is reduced, we may assume that fr−1 = x · gr−1 and fr = y · gr with gr−1 , gr

2.3 Gamma-invariant

49

convenient. Hence γ(f ) = γ(f1 · . . . · fr−2 · gr−1 gr · xy) = γ(xy) + γ(f1 · . . . · fr−2 · gr−1 gr ) + 2 · i(xy, f1 · . . . · fr−2 · gr−1 gr ) − 1 r−2 X X = 1+ γ(fi ) + i(fi , fj ) + γ(gr−1 ) + γ(gr ) + 2 · i(gr−1 , gr ) i=1

+

r−2 X

j6=i

2 · i(gr−1 gr , fi ) − r + 1

i=1

+2 i(xy, f1 · . . . · fr−2 ) + i(x, gr−1 ) + i(y, gr−1 ) + i(x, gr ) + i(y, gr ) − 1 r−2 X X = 1+ γ(fi ) + i(fi , fj ) + γ(fr−1 ) − 1 + γ(fr ) − 1 + 2 · i(fr−1 , fr ) i=1

+

r−2 X

j6=i

2 · i(fr−1 fr , fi ) − r

i=1

=

r X i=1

γ(fi ) +

X

i(fi , fj ) − r + 1.

j6=i

Definition 2.3.3. Let char(K) = p ≥ 0 and let f = f1 · . . . · fr ∈ K[[x, y]] be reduced with fi irreducible. 1. We call p to be multiplicity good (m-good) for f if p does not divide the multiplicity mt(fi ) for all i = 1, . . . , r. 2. p is called intersection multiplicity good (im-good) for f if for all i = 1, . . . , r, either p - i(fi , x) or p - i(fi , y). Note that “m-good” implies “im-good”. Lemma 2.3.4. Let f ∈ K[[x, y]] be reduced such that p is m-good for f . Then for every g ∼c f , we have γ(g) = γ(f ). Proof. It suffices to show that γ(f ) ≥ γ(g). Moreover since γ(f ) = γ(u · f ) for every unit u we may assume that g ∼r f . This implies that m := mt(f ) = mt(g) and there exists a coordinate change Φ : K[[x, y]] → K[[x, y]] x 7→ u(x, y) y 7→ v(x, y) such that f (x, y) = g(u(x, y), v(x, y)). Take a parametrization (x(t)|y(t)) of f , then u(t) := u(x(t), y(t)) v(t)) := v(x(t), y(t))

50


is a parametrization of g. Without loss of generality we may assume that ordx(t) ≤ ordy(t) and that ordu(t) ≤ ordv(t). Then ordx(t) = ordu(t) = m and therefore ordx0 (t) = ordu0 (t) = m − 1 by assumption that p does not divide m. Since f (x(t), y(t)) = 0, fx (x(t), y(t)) · x0 (t) + fy (x(t), y(t)) · y 0 (t) = 0. Therefore, i(f, fx ) + ordx0 (t) = i(f, fy ) + ordy 0 (t). It yields i(f, fx ) + ordx(t) ≥ i(f, fy ) + ordy(t) since ordx0 (t) = ordx(t) − 1 = m − 1 and since the fact that ordy 0 (t) ≥ ordy(t) − 1. This implies that γ(f ) = i(f, fy ) − m + 1 and i(f, fy ) ≤ i(f, fx ). Analogously, we have γ(g) = i(g, gv ) − m + 1 and i(g, gv ) ≤ i(g, gu ). Besides, since f (x, y) = g(u(x, y), v(x, y)), fy = gu · uy + gv · vy . Then fy (x(t), y(t)) = gu (u(t), v(t)) · uy (x(t), y(t)) + gv (u(t), v(t)) · vy (x(t), y(t)). This implies that i(f, fy ) ≥ min{i(g, gu ) + orduy (x(t), y(t)), i(g, gv ) + ordvy (x(t), y(t))} ≥ min{i(g, gu ), i(g, gv )} = i(g, gv ). Hence γ(f ) = i(f, fy ) − m + 1 ≥ i(g, gv ) − m + 1 = γ(g).

Lemma 2.3.5. Let f ∈ K[[x, y]] be irreducible such that m = i(f, x) = i(f, y). Let g ∈ K[[x, y]] be such that f (x, y) = g(x, αx − βy), where (β : α) ∈ KP1 is the unique tangent direction of f . Then (i) m = i(g, x) < i(g, y). (ii) γ(f ) ≥ γ(g). (iii) If the characteristic p of K is im-good for g but not for f , then γ(f ) > γ(g).

2.3 Gamma-invariant

51

Proof. (i) Let f = fm + fm+1 + . . . be the homogeneous decomposition of f . Since i(f, x) = i(f, y), m = mt(f ) = i(f, x) = i(f, y). Then by , fm = (αx − βy)m ; α, β 6= 0. It follows from Proposition 1.2.19 that i(g, x) = i g(x, αx − βy), x = i(f, x) = m and n := i(g, y) = i g(x, αx − βy), αx − βy = i(f, αx − βy) > m. (ii) Let (x(t), y(t)) be a parametrization of f . Then X(t) = x(t) Y (t) = αx(t) − βy(t) is a parametrization of g. Since f (x, y) = g(x, αx − βy), fx (x, y) = gx (x, αx − βy) − αgy (x, αx − βy) fy (x, y) = βgy (x, αx − βy) and therefore fx (x(t), y(t)) = gx (X(t), Y (t)) − αgy (X(t), Y (t)) fy (x(t), y(t)) = βgy (X(t), Y (t)). We consider two following cases: – If i(f, fx ) ≥ i(f, fy ). Then by Lemma 2.3.2, γ(f ) = = = ≥

min{i(f, fx ) − i(f, y) + 1, i(f, fy ) − i(f, x) + 1} i(f, fy ) − m + 1 i(g, gy ) − i(g, x) + 1 γ(g).

– If i(f, fx ) < i(f, fy ), then ordfx (x(t), y(t)) < ordfy (x(t), y(t)) = gy (X(t), Y (t)). This, together with the equality fx (x(t), y(t)) = gx (X(t), Y (t)) − αgy (X(t), Y (t)) imply that ordfx (x(t), y(t)) = ordgx (X(t), Y (t)) < ordgy (X(t), Y (t)), or equivalently i(f, fx ) = i(g, gx ) < i(g, gy ). It follows from Lemma 2.3.2 that γ(g) = = < = =

min{i(g, gx ) − i(g, y) + 1, i(g, gy ) − i(g, x) + 1} i(g, gx ) − i(g, y) + 1 i(f, fx ) − i(g, x) + 1 i(f, fx ) − m + 1 γ(f ).

52


(iii) As in the proof of part (ii), if i(f, fx ) < i(f, fy ) then γ(f ) > γ(g). Assume now that i(f, fx ) ≥ i(f, fy ). Then as above, we have γ(f ) = i(g, gy ) − i(g, x) + 1. Since p is not im-good for f , p | m and therefore p - n since p is im-good for g. This, together with ord Y (t) = i(g, y) = n and ord X(t) = i(g, x) = m implies that ˙ ord Y˙ (t) = n − 1 = i(g, y) − 1 and ord X(t) > m − 1 = i(g, x) − 1. On the other hand, since g(X(t), Y (t)) = 0, we have ˙ X(t) · gx (X(t), Y (t)) + Y˙ (t) · gy (X(t), Y (t)) = 0. It yields ˙ ord X(t) + ord gx (X(t), Y (t)) = ord Y˙ (t) + ord gy (X(t), Y (t)), or ˙ i(g, gx ) − ord Y˙ (t) = i(g, gy ) − ord X(t). This implies that i(g, gx ) − i(g, y) < i(g, gy ) − i(g, x). Hence γ(g) = = < = =

min{i(g, gx ) − i(g, y) + 1, i(g, gy ) − i(g, x) + 1} i(g, gx ) − i(g, y) + 1 i(g, gy ) − i(g, x) + 1 i(f, fx ) − m + 1 γ(f ).

Lemma 2.3.6. Let f ∈ K[[x, y]] be irreducible and f˜ its strict transform (cf. Section 1.2.4), then γ(f ) ≥ m2 − m + γ(f˜). If i(f, x) 6= i(f, y) then (i) γ(f ) = m2 − m + γ(f˜), with m := mt(f ) the multiplicity of f . (ii) p is im-good for f , if and only if it is so for f˜. Proof. (i) If f is not convenient then either f = x · u or f = y · u for some unit u since f is irreducible and hence the lemma is evident. Assume now that f is convenient and that i(f, x) < i(f, y). Then the (local equation of) f˜ at the point (1 : 0) in the first chart is: f (u, uv) = um f˜(u, v)

2.3 Gamma-invariant

53

and therefore fx (u, uv) + vfy (u, uv) = mum−1 f˜(u, v) + um f˜u (u, v) ufy (u, uv) = um f˜v (u, v). It yields xfx (x, y) + yfy (x, y) = mum f˜(u, v) + um uf˜u (u, v) yfy (x, y) = um v f˜v (u, v) ,

where x = u, y = uv. Take a parametrization (u(t)|v(t)) of f˜. Then x(t) = u(t) y(t) = u(t)v(t)) will be a parametrization of f and x(t)fx (x(t), y(t)) + y(t)fy (x(t), y(t)) = u(t)m u(t)f˜u (u(t), v(t)) y(t)fy (x(t), y(t)) = u(t)m v(t)f˜v (u(t), v(t)) . Thus αx(t)fx (x(t), y(t))+(α+β)y(t)fy (x(t), y(t)) = αu(t)f˜u (u(t), v(t))+βv(t)f˜v (u(t), v(t)), for (α : β) ∈ KP1 generic. It follows that i f, αxfx + (α + β)yfy = m2 + i f˜, αuf˜u + βv f˜v since ord u(t) = m. Besides, i(f, x) + i(f, y) = = = =

ordx(t) + ordy(t) ordu(t) + ordu(t) + ordv(t) m + ordu(t) + ordv(t) m + i(f˜, u) + i(f˜, v).

Hence by definition we have γ(f ) = i(f, αxfx + (α + β)yfy ) − i(f, x) − i(f, y) + 1 = m2 − m + i(f˜, αuf˜u + βv f˜v ) − i(f˜, u) − i(f˜, v) + 1 = m2 − m + γ(f˜). (ii) follows from the equalities i(f, x) = ord x(t) = ord u(t) = i(f˜, u) and i(f, y) = ord y(t) = ord u(t) + ord v(t) = i(f˜, u) + i(f˜, v).

54

Chapter 2. Plane Curve Singularities In general, it is sufficient to prove γ(f ) ≥ m2 − m + γ(f˜) for provided i(f, x) = i(f, y). Let (β : α) be the unique tangent direction of f and g ∈ K[[x, y]] such that f (x, y) = g(x, αx − βy). Then by Lemma 2.3.5, i(g, x) < i(g, y) and γ(f ) ≥ γ(g). It follows from Part (i) that γ(g) ≥ m2 − m + γ(˜ g ), where g˜ is the strict transform of g. Besides, it is easy to see that the local equation of f˜ at the point (β : α) coincides that of g˜ at the point (1 : 0). This means that γ(f˜) = γ(˜ g ). Hence γ(f ) ≥ γ(g) ≥ m2 − m + γ(˜ g ) = m2 − m + γ(f˜).

Proposition 2.3.7. Let f ∈ K[[x, y]] be reduced. Then γ(f ) ≥ 2δ(f ) − r(f ) + 1. Equality holds if and only if the characteristic p is im-good for f . Proof. The proof will be divided into two steps Step 1: f is irreducible. We argue by induction on the delta-invariant of f . If δ(f ) = 0, i.e. f is a nonsingular and then γ(f ) = 0. Suppose that δ(f ) > 0. Then m > 1. It follows from Proposition 1.2.31 that δ(f ) =

m(m − 1) + δ(f˜) > δ(f˜). 2

Combining the induction hypothesis, Lemma 2.3.6 and Proposition 1.2.31, we obtain γ(f ) ≥ m2 − m + γ(f˜) ≥ m2 − m + 2δ(f˜) = 2δ(f ). Assume now that p is im-good for f . • If i(f, x) 6= i(f, y) then γ(f ) = γ(f˜) and p is also im-good for f˜ by Lemma 2.3.6. By induction hypothesis, γ(f˜) = 2δ(f˜). It hence follows from Lemma 2.3.6 and Proposition 1.2.31 that γ(f ) = m2 − m + γ(f˜) = m2 − m + 2δ(f˜) = 2δ(f ). • If i(f, x) = i(f, y), then i(f, x) = i(f, y) = m and therefore p - m by assumption that p is im-good for f . Take g ∈ K[[x, y]] as in Lemma 2.3.5 then

2.3 Gamma-invariant

55

γ(f ) = γ(g) by Lemma 2.3.4 and δ(f ) = δ(g) by Proposition 1.2.19. Applying induction hypothesis to the strict transform g˜ of g gives γ(˜ g ) = 2δ(˜ g ). Combining Lemma 2.3.6 with Proposition 1.2.31 we get γ(f ) = γ(g) = m2 − m + γ(˜ g) 2 = m − m + 2δ(˜ g) = 2δ(g) = 2δ(f ). Finally, we will prove that γ(f ) > 2δ(f ) if p is not im-good for f by induction on the delta-invariant of f . Since p is not im-good for f , mt(f ) ≥ p and hence δ(f ) ≥ p(p − 1)/2. If δ(f ) = p(p − 1)/2, then mt(f ) = p and f becomes non-singular after one blowing up. We may write f = fp + fp+1 + . . . , where fp = (αx − βy)p with β 6= 0. We will show that α 6= 0. By contradiction, suppose that α = 0. Then i(f, y) > p = i(f, x). Besides p | i(f, y) since p is not im-good for f and therefore i(f, y) ≥ 2p. Thus it is easy to see that the multiplicity of the strict transform of f is greater than or equal to p, which contradicts the fact that f becomes non-singular after one blowing up. Since α 6= 0, i(f, y) = i(f, x) = p. Taking g ∈ K[[x, y]] as in Lemma 2.3.5 yields p = i(g, x) < i(g, y). On the other hand, g˜ must be non-singular since p(p − 1)/2 = δ(f ) = δ(g) = p(p − 1)/2 + δ(˜ g ). This implies that i(˜ g , v) = 1. Hence i(g, y) = i(˜ g , v) + i(g, x) = p + 1. Consequently, p is im-good for g and therefore γ(f ) > γ(g) by Lemma 2.3.5. Applying the first part to g we have γ(g) ≥ 2δ(g) and hence γ(f ) > γ(g) ≥ 2δ(g) = 2δ(f ). Now we prove the induction step. Suppose that δ(f ) > p(p−1)/2. If i(f, x) 6= i(f, y) then p is not im-good for f˜ by Lemma 2.3.6 since it is not im-good for f . We can apply the induction hypothesis to f˜ and obtain γ(f ) = m(m − 1) + γ(f˜) > m(m − 1) + 2δ(f˜) = 2δ(f ). If i(f, x) = i(f, y). Take g ∈ K[[x, y]] as in Lemma 2.3.5. If p is not im-good for g, we now apply the above argument, with f replaced by g, to obtain γ(g) > 2δ(g) and hence γ(f ) = γ(g) > 2δ(g) = 2δ(f )

56


by Lemma 2.3.5 and Proposition 1.2.19. This proves Step 1. Step 2: Assume that f decomposes into its branches f = f1 · . . . · fr . Then γ(f ) =

r X

γ(fi ) +

i=1

and 2δ(f ) =

X

i(fi , fj ) − r + 1

j6=i

r X

2δ(fi ) +

i=1

X

i(fi , fj ) .

j6=i

The proposition follows from the above equalities and Step 1. Corollary 2.3.8. Let f ∈ K[[x, y]] be such that p := char(K) is m-good for f . Then 2δ(f ) = i(f, αfx + βfy ) − mt(f ) + r(f ) with (α : β) ∈ KP1 generic. In particular if m := mt(f ) = ordf (0, y) then 2δ(f ) = i(f, fy ) − m + r. Proof. Step 1: Assume first that f is irreducible. Let (x(t), y(t)) be a parametrization of f . Then i(f, αfx + βfy ) = ord αfx (x(t), y(t)) + βfy (x(t), y(t)) = min{ord fx (x(t), y(t)); ord fy (x(t), y(t))} = min{i(f, fx ); i(f, fy )} Without loss of generality we may assume that ord x(t) ≤ ord y(t). Using the same argument as in the proof of Lemma 2.3.4 we obtain that γ(f ) = i(f, fy ) − mt(f ) + 1 and i(f, fy ) ≤ i(f, fx ). It yields that γ(f ) = i(f, fy ) − mt(f ) + 1 = i(f, αfx + βfy ) − mt(f ) + 1. By Proposition 2.3.7, 2δ = γ(f ) = i(f, αfx + βfy ) − mt(f ) + 1. Step 2: Now assume that f decomposes as f = f1 · . . . · fr with fi irreducible. Then i(f, αfx + βfy ) = =

r X i=1 r X

i fi , α

∂fi X ∂fi +β + i(fi , fj ) ∂x ∂y j6=i

2δ(fi ) + mt(fi ) − 1 +

i=1

= 2δ(f ) + mt(f ) − r, due to the first step, respectively Proposition 1.2.14.

X j6=i

i(fi , fj )

2.3 Gamma-invariant

57

Proposition 2.3.9. Let f ∈ K[[x, y]] be irreducible such that m := mt(f ) = ordf (0, y). Assume that p := char(K) is m-good for f . Let x(t) = tm y(t) = a1 tn1 + a2 tn2 + . . . be a parametrization of f , where m ≤ n1 < n2 < . . . and ai 6= 0 for all i ≥ 1. Then 1X δ(f ) = (ni − 1)(Di−1 − Di ), 2 i≥1 where D0 := m and Di := gcd(m, n1 , . . . , ni ) for i ≥ 1. Proof. By Corollary 2.3.8, 2δ(f ) = i(f, fy ) − m + 1. We shall describe the intersection multiplicity i(f, fy ) in terms of the parametrization (x(t)|y(t)). By Proposition 1.2.6, there exists a unit u such that f =u·

m Y

y − y(ξ j x1/m ) ,

j=1

where ξ is a primitive m-th root of unity. Then fy = uy ·

m Y

j 1/m

y − y(ξ x

m Y X ) +u· y − y(ξ k x1/m )

j=1

j=1 k6=j

and therefore fy (x(t), y(t)) =

Y

y(t) − y(ξ k t)

k6=m

since

Q

k6=j

y(t) − y(ξ k t) = 0 for all j = 1, . . . , m − 1. This implies i(f, fy ) =

m−1 X

ord y(t) − y(ξ j t) .

j=1

We divide the set {1, . . . , m − 1} into disjoint subsets I1 , I2 , . . . , IN := ∅ defined by Ii := {j ∈ {1, . . . , m − 1}|jDi−1 ∈ mZ, jDi 6∈ mZ}, 1 ≤ i ≤ N. It is easy to see that the cardinality ]Ii = Di−1 − Di . So we can write i(f, fy ) =

N X X

ord y(t) − y(ξ j t) .

i=1 j∈Ii

Fix i ≥ 1, for each j ∈ Ii and for every k < i, we have jnk ∈ mZ since jDi−1 ∈ mZ and Di−1 divides nk , and therefore ξ jnk = 1. While ξ jni 6= 1 since jni 6∈ mZ. This implies that y(t) − y(ξ j t) = ai (1 − ξ jni )tni + terms of higher order.

58


Consequently, ord y(t) − y(ξ j t) = ni , for all j ∈ Ii . It follows that i(f, fy ) =

N X

(Di−1 − Di )ni .

i=1

Hence 2δ(f ) = i(f, fy ) − m + 1 = =

N X i=1 N X

(Di−1 − Di )ni −

N X

(Di−1 − Di )

i=1

(ni − 1)(Di−1 − Di ).

i=1

This proves the proposition.

2.4

Milnor formula

Definition 2.4.1. Let ϕ(t) ∈ K[[t]]. Let f, g ∈ K[[x, y]] and g = g1 · · · gs with gi irreducible. Let t → 7 (xi (t), yi (t)) be a parametrization of gi for i = 1, . . . , s. We define dϕ • dexp(ϕ) := ord the differential exponent of ϕ. dt • decart(ϕ) := dexp(ϕ) − ordϕ + 1 the differential ecart of ϕ. P • decart(f |g) := i decart(f (xi (t), yi (t))) the differential ecart of f restricted on g. For each (α : β) ∈ KP1 , we denote • Pα:β (f ) = αfx + βfy the (α : β)-polar of f and Lα:β = αx + βy. • (α : β)−pcd(f ) = decart(f |Pα:β (f ))−decart(f |L−β:α )−decart(L−β:α |Pα:β (f )). • pcd(f ) = (α : β)-pcd(f ) with (α : β) generic, the polar characteristic difference of f . Remark 2.4.2. 1. If the characteristic p of K does not divide ordϕ, then decart(ϕ) = 0. 2. If charK = 0, then decart(ϕ) = 0 for any ϕ(t) ∈ K[[t]] and hence pcd(f ) = 0 for any f ∈ K[[x, y]]. 3. The differential ecart is not symmetric, i.e. decart(f |g) 6= decart(g|f ) in general. E.g. f = x2 − y 5 and g = (x − y)3 − y 5 are both irreducible. (t5 , t2 ) and (t3 + t5 , t3 ) are respectively parametrizations of f, g. Then f (t3 + t5 , t3 ) = (t3 + t5 )2 − (t3 )5 = t6 + 2t8 + t10 − t15

2.4 Milnor formula

59

and g(t5 , t2 ) = (t5 − t2 )3 − (t2 )5 = −t6 + t15 − t10 . Hence decart(f |g) = 2 6= decart(f |g) = 4. 4. The (α : β)−pcd is not constant in general. E.g. f (x, y) = x2 + y 3 + y 5 ∈ K[[x, y]], with char(K) = 3. Then If (α : β) = (1 : 0), then decart(f |fx ) = 2, decart(y|fx ) = 0, decart(f |y) = 0 =⇒ (1 : 0)-pcd(f ) = 2. If (α : β) = (0 : 1), then decart(f |fy ) = 0, decart(x|fy ) = 0, decart(f |x) = 2 =⇒ (0 : 1)-pcd(f ) = −2. Proposition 2.4.3. For any (α : β) ∈ KP1 , we have µ(f ) = i(f, Pα:β (f )) − i(−βx + αy, Pα:β (f ))) + (α : β)-pcd(f ) + decart(f |L−β:α ) = i(f, Pα:β (f )) − i(f, −βx + αy) + (α : β)-pcd(f ) + 1. Proof. It suffices to prove the proposition for (α : β) = (1 : 0), i.e. we have to show µ(f ) = i(f, fx ) − i(fx , y) + (1 : 0)-pcd(f ) + decart(f |y) = i(f, fx ) − i(f, y) + (1 : 0)-pcd(f ) + 1. Let fx = g1 · · · gs be an irreducible decomposition of fx . Let t 7→ (xi (t), yi (t)) be a parametrization of gi for i = 1, . . . , s. Then

= = = =

ordf (xi (t), yi (t)) d ord( f (xi (t), yi (t))) + 1 − decart(f (xi (t), yi (t))) dt d ord fy (xi (t), yi (t)) yi (t) + 1 − decart(f (xi (t), yi (t))) dt d ordfy (xi (t), yi (t)) + ord yi (t) + 1 − decart(f (xi (t), yi (t))) dt ordfy (xi (t), yi (t)) + ordyi (t) + decart(yi (t)) − decart(f (xi (t), yi (t))).

Hence i(f, fx ) = i(fx , fy ) + i(fx , y) + decart(y|fx ) − decart(f |fx ) = µ(f ) + i(fx , y) − (1 : 0)-pcd(f ) − decart(f |y). This prove the first equation of the proposition. The second follows from the first and the fact i(fx , y) = ordx (fx (x, 0)) = decart(f (x, 0)) + ordx (f (x, 0)) − 1 = decart(f |y) + i(f, y) − 1.

60


Theorem 2.4.4. Let f ∈ K[[x, y]] be such that p is m-good for f . Then Sw(f ) = µ(f ) − (2δ(f ) − r(f ) + 1) = pcd(f ). Proof. The theorem follows from Proposition 2.4.3, Corollary 2.3.8 and the fact that i(f, −βx + αy) = mt(f ) with (α : β) generic.

2.5

Parametrization determinacy

As known in [BGM12] that a singularity f ∈ K[[x1 , . . . , xn ]] is finitely right (resp. contact) determined if and only if it is an isolated singularity (resp. isolated hypersurface singularity) and then they gave also an upper bound for the determincy in terms of the Milnor number (resp. Tjurina number) of f . We prove that every reduced plane curve singularity f is finitely parametrization determined. Moreover we show that the determinacy of f is at most c + 1, where c is its conductor.   x1 (t) y1 (t) L   .. Definition 2.5.1. Let R0 = ri=1 K[[t]]. Let ψ =  ...  ∈ R02 . For each . xr (t) yr (t) r k = (k1 , . . . , kr ) ∈ N , we call   j k1 x1 (t) j k1 y1 (t)   .. j k ψ :=  ...  . j kr xr (t) j kr yr (t)

the k-jet of ψ, where j ki xi (t) (resp. j ki yi (t)) is the ki -jet of xi ∈ K[[t]] (resp. yi ∈ K[[t]]). Definition 2.5.2. Let f ∈ K[[x, y]] be reduced, let ψ : K[[x, y]] → R0 =

r M

K[[t]]

i=1

its parametrization and let k = (k1 , . . . , kr ) ∈ Nr . We call f (or ψ) parametrization k-determined if ψ is parametrization equivalent every ψ 0 whose k-jet coincides with that of ψ. We say that f is parametrization finitely determined if it is parametrization k-determined for some k = (k1 , . . . , kr ) ∈ Nr . Theorem 2.5.3 (Parametrization determinacy of plane curve singularities). Let f ∈ K[[x, y]] be reduced, r the number of the irreducible components, c ∈ Zr+ its conductor. • If mt(f ) = 1, then f is parametrization 1-determined. • If mt(f ) = 2 and r = 1 then f is parametrization c + 1-determined. • If mt(f ) = 2 and r = 2 then f is parametrization c-determined.

2.5 Parametrization determinacy

61

• If mt(f ) > 2, then f is parametrization (c − 1)-determined, where 1 = (1, . . . , 1) ∈ Zr . In particular, f is always parametrization (c + 1)-determined. For the proof we need the two following lemmas, which give several relations between the delta-invariant (δ), the conductor (c) and the maximal contact multiplicity (β¯1 ) of a reduced power series in some concrete cases. Lemma 2.5.4. Let f = f1 ·f2 ∈ K[[x, y]] be reduced such that f1 , f2 are non-singular. Then β¯1 (f ) = i(f1 , f2 ) = δ(f ). Proof. The second equality follows from Proposition 1.2.14. We now prove the first one. By definition, β¯1 (f ) ≥ min{i(f1 , f1 ), i(f1 , f2 )} = i(f1 , f2 ). It remains to prove min{i(f1 , γ), i(f1 , γ)} ≤ i(f1 , f2 ), for every non-singular series γ. Since γ is non-singular, Lemma 1.2.18 yields that there exists a coordinate change Φ ∈ AutK K[[x, y]] such that Φ(γ) = y. By Proposition 1.2.19 it suffices to show min{i(F1 , y), i(F2 , y)} ≤ i(F1 , F2 ), where F1 := Φ(f1 ) and F2 := Φ(f2 ). Since i(F1 , F2 ) ≥ 1, we may assume that k := i(F1 , y) > 1 and l := i(F2 , y) > 1. Then X aij xi y j ; a01 , ak0 6= 0 F1 (x, y) = a01 y + ak0 xk + i+kj>k

and F2 (x, y) = c01 y + cl0 xl +

X

cij xi y j ; c01 , cl0 6= 0.

i+lj>l

Thus F1 has a parametrization x(t) = t; y(t) = atk + terms of higher order. Therefore F2 (x(t), y(t)) = ac01 tk + terms of higher order + cl0 tl + terms of higher order. Hence i(F1 , F2 ) = ord F2 (x(t), y(t)) ≥ min{k, l} = min{i(F1 , y), i(F2 , y)}.

Lemma 2.5.5. Let f ∈ K[[x, y]] be irreducible. (i) If mt(f ) = 2, then c(f ) = 2δ(f ) = β¯1 (f ) − 1. (ii) If mt(f ) > 2, then c(f ) > β¯1 (f ).

62


Proof. By using Lemma 1.2.18 and Proposition 1.2.19, it suffices to prove the lemma in the case m - n, where m = mt(f ) = ordf (0, y) and n = ordf (x, 0). Note that in this case β¯1 (f ) = n. By Proposition 1.2.21, the Newton polygon Γ(f ) is the unique edge E which joints the points A = (m, 0) and B = (0, n). Then we can easily see that (m − 1)(n − 1) . δN (f ) = 2 (i) If m = mt(f ) = 2 then n is odd and fE = c20 y 2 + c0n xn . Clearly f is WNND, it follows from Proposition 2.2.10 that δ(f ) = δN (f ). Hence c(f ) = 2δ(f ) = 2δN (f ) = n − 1 = β¯1 (f ) − 1. (ii) If m = mt(f ) > 2 then n > m ≥ 3. By Proposition 2.2.10, δ(f ) ≥ δN (f ). Hence c(f ) = 2δ(f ) ≥ 2δN (f ) = (m − 1)(n − 1) > n = β¯1 (f ).

Proof of Theorem 2.5.3. The first statement is evident. We put now   c + 1 if mt(f ) = 2 and r = 1 r Z≥0 3 k := c if mt(f ) = 2 and r = 2   c − 1 if mt(f ) > 2. Note that k ≥ c − 1, i.e. ki ≥ ci − 1 parametrization k-determined. x1 (t)  .. Let ψ(t) := (x(t)|y(t)) =  .

for all i = 1, . . . , r. We will show that f is

 y1 (t)  ..  be a parametrization of f and let . x (t) yr (t) Lr r 0 0 0 ψ (t) = (x (t)|y (t)) ∈ R0 = i=1 K[[t]] such that j k (ψ) = j k (ψ 0 ). Then x(t) − x0 (t) ∈ tk+1 R0 ⊂ R and y(t) − y 0 (t) ∈ tk+1 R0 ⊂ R, where R = K[[x, y]]/hf i. Thus there exist g1 , g2 ∈ K[[x, y]] such that g1 (ψ(t)) = x(t) − x0 (t) ∈ tk+1 R0 and g2 (ψ(t)) = y(t) − y 0 (t) ∈ tk+1 R0 . This implies g1 (xi (t), yi (t)), g2 (xi (t), yi (t)) ∈ tki +1 K[[t]], ∀i = 1, . . . , r. Claim 4. mt(g1 ) > 1 (similarly, mt(g2 ) > 1).


63

Proof. We argue by contradiction. Suppose that it is not true, i.e. mt(g1 ) = 1. Then by definition, min{i(fi , g1 )|i = 1, . . . , r} ≤ β¯1 (f ), where β¯1 (f ) denotes the maximal contact multiplicity of f . We consider three following cases: Case 1: mt(f ) = 2 and r = 1. Then k = c + 1 and g1 (ψ(t)) ∈ tk+1 K[[t]]. This implies i(f, g1 ) = ord g1 (ψ(t)) ≥ k + 1 = c + 2. By Lemma 2.5.5, i(f, g1 ) ≥ c + 2 = β¯1 (f ) + 1, which is a contradiction. Case 2: mt(f ) = 2 and r = 2. Then f = f1 · f2 with mt(f1 ) = mt(f2 ) = 1 and k = c. It follows from Propsition 1.2.15 that c1 = k1 = c2 = k2 = i(f1 , f2 ). Therefore i(g1 , f1 ) = ord g1 (x1 (t), y1 (t)) ≥ k1 + 1 = i(f1 , f2 ) + 1, since g1 (x1 (t), y1 (t)) ∈ tk1 +1 K[[t]]. Similarly, i(g1 , f2 ) ≥ i(f1 , f2 )+1. Then by Lemma 2.5.4, i(f1 , f2 ) + 1 ≤ min{i(f1 , g1 ); i(f2 , g1 )} ≤ β¯1 (f ) = i(f1 , f2 ), a contradiction. Case 3: mt(f ) > 2. Then k = c − 1. Let f = f1 · . . . · fr be an irreducible decomposition of f such that mt(f1 ) ≤ . . . ≤ mt(fr ). • If mt(fr ) > 2, then i(fr , g1 ) = ord g1 (xr (t), yr (t)) ≥ kr + 1 = cr . By Lemma 2.5.5 and by definition, c(fr ) > β¯1 (fr ) ≥ i(fr , g1 ) ≥ cr > c(fr ), a contradiction. • If mt(fr ) = 2, then r > 1 and i(fr , g1 ) = ord g1 (xr (t), yr (t)) ≥ kr + 1 = cr . This implies that β¯1 (fr ) ≥ cr . i(f1 , fr ) ≥ mt(fr ) = 2,

By Propsition 1.2.15 and the inequality

cr ≥ c(fr ) + i(f1 , fr ) > c(fr ) + 1. Hence by Lemma 2.5.5, c(fr ) = β¯1 (fr ) − 1 ≥ cr − 1 > c(fr ), which is a contradiction.

64

Chapter 2. Plane Curve Singularities • If mt(fr ) = 1 then r > 2 and mt(f1 ) = mt(f2 ) = 1 since the assumption that mt(f1 ) ≤ . . . ≤ mt(fr ). Propsition 1.2.15 yields that c1 ≥ i(f1 , f2 ) + i(f1 , fr ) ≥ i(f1 , f2 ) + 1. Thus i(f1 , g1 ) = ord g1 (x1 (t), y1 (t)) ≥ k1 + 1 = c1 ≥ i(f1 , f2 ) + 1. Similarly i(f2 , g1 ) ≥ i(f1 , f2 )+1 and then i(f1 , f2 )+1 ≤ min{i(f1 , g1 ); i(f2 , g1 )}. It hence follows from Lemma 2.5.4 that i(f1 , f2 ) + 1 ≤ min{i(f1 , g1 ); i(f2 , g1 )} ≤ β¯1 (f1 · f2 ) = i(f1 , f2 ), a contradiction.

Now, applying the above claim, the following map Φ : K[[x, y]] −→ K[[x, y]] x 7→ x − g1 (x, y) y 7→ y − g2 (x, y) is an automorphism of K[[x, y]] and ψ ◦ Φ(x) = ψ(x − g1 (x, y)) = x(t) − g1 (ψ(t)) = x0 (t) and ψ ◦ Φ(y) = ψ(y − g2 (x, y)) = y(t) − g2 (ψ(t)) = y 0 (t). This implies that ψ ◦ Φ = ψ 0 , i.e. ψ ∼p ψ 0 . Example 2.5.6. 1. Let f = x2 − y 5 . Then r(f ) = 1, c(f ) = 4 and (t5 , t2 ) is a parametrization of f . It is easy to see that f is not parametrization 4determined. 3 3 t t 3 5 2. Let f = (x − y )(x − y ). Then r(f ) = 2, c(f ) = and is a 3 t5 t parametrization of f . It can be easily checked that 3 3 t t t t 0 t ∼p 6∼p . t5 t 0 t 0 t This means that f is not parametrization

2 2

-determined.

Corollary 2.5.7. Let char(K) 6= 2 and let f ∈ K[[x, y]] be irreducible with mt(f ) = 2 and δ(f ) = k ≥ 1. Then f is contact equivalent to x2k+1 − y 2


65

Proof. Since mt(f ) = 2, it follows from Lemma 1.2.18 that f is contact equivalent to g with ordg(0, y) = 2, ordg(x, 0) = m, 2 - m and β¯1 (f ) = β¯1 (g) = m. By Lemma 2.5.5(i), m = β¯1 (f ) = 2δ(f ) + 1 = 2k + 1. Let ψ = (t2 , y(t)) be the Puiseux parametrization of g. Then ordy(t) = 2k + 1 due to Proposition 1.2.21. Theorem 2.5.3 yields that ψ ∼ (t2 , a · t2k+1 ) for some a 6= 0. This implies that ψ ∼ (t2 , a · t2k+1 ) ∼ (t2 , t2k+1 ) by coordinate change x 7→ x, y 7→ y/a and hence f ∼c x2k+1 − y 2 .

Chapter 3 Modality In the sixties V. I. Arnol’d introduced in [AGV85, Part II]) the notion of modality as follows: “the modality r of a point x ∈ X under the action of a Lie group G on a manifold X is the least number such that a sufficiently small neighbourhood of x may be covered by a finite number of r-parameter families of orbits. The modality of a function-germ at a critical point with critical value 0 is defined to be the modality of a sufficient jet in the space of jets of functions with critical point 0 and critical value 0.” And he classified simple and unimodal singularities (i.e. r = 0, 1) under the action of right grouo (i.e. with respect to right equivalence). We generalize this notion to algebraic setting and moreover to an equivalence relation induced by a morphism h : X 0 → X. This allows us to relate modality with formal deformation theory. More precise, we show that the semi-universal modality is sufficient to determine the modality of a given singularity. The results of this chapter are used for the classification in Chapter 4. We first recall some of the classical results in dimension theory [Har77], [Mum88], [Spr81].

3.1

The fibers of a morphism

Theorem 3.1.1. Let f : X → Y be a dominant morphism of irreducible algebraic varieties 1 and let r = dim X − dim Y . Let W ⊂ Y be a closed irreducible subset and let Z be an irreducible component of f −1 (W ). (1) If Z dominates W then dim Z ≥ dim W + r. In particular, for y ∈ f (X), any irreducible component of f −1 (y) has dimension ≥ r. (2) There is an open dense subset U ⊂ Y (depending only on f ) such that U ⊂ f (X) and dim Z = dim W + r or Z ∩ f −1 (U ) = ∅. In particular, for y ∈ U , any irreducible component of f −1 (y) has dimension equal to r. (3) If X and Y are affine, then the open set U in (2) may be chosen such that 1

By an algebraic variety we mean a separated scheme of finite type over an algebraically closed field K. By a point we mean a closed point, see [Har77].

67

68

Chapter 3. Modality f : f −1 (U ) → U factors as follows f −1 (U )

ϕ

/

U × Kr .

f

&

pr1

U

with π finite and pr1 the projection onto the first factor. Proof. cf. [Mum88, I. 8] and [Spr81, Thm. 4.1.6]. Corollary 3.1.2 (Chevalley). Let f : X → Y be a morphism of irreducible algebraic varieties. Then the image of f is a constructible set 2 in Y . More generally, f maps constructible sets in X to constructible sets in Y . Proof. cf. [Mum88, I. 8. Coro. 2]. Corollary 3.1.3. Let f : X → Y be a dominant morphism of irreducible algebraic varieties such that dim f −1 (f (x)) = i for all x ∈ X and for some i ∈ N. Then dim X = i + dim Y. Proof. Straight forward from Theorem 3.1.1 (2). Corollary 3.1.4. Let f : X → Y be a morphism algebraic varieties such that dim f −1 (f (x)) = i for all x ∈ X and for some i ∈ N. Then dim X ≤ i + dim Y. Proof. We may assume that f is surjective. Let X1 , . . . , Xk be the irreducible components of X and let Yj = f (Xj ). Then the restrictions f j : X j → Yj are dominant morphism of irreducible algebraic varieties. By Theorem 3.1.1 (1), for y ∈ f (Xj ) we have dim Xj − dim Yj ≤ dim fj−1 (y). Hence dim Xj − dim Yj ≤ dim f −1 (y) = i. Taking j = 1, . . . , k such that dim X = dim Xj we get dim X = dim Xj ≤ dim Yj + i ≤ dim Y + i, which completes the proof. 2

By a constructible set we mean a union of locally closed subsets

3.1 The fibers of a morphism

69

Corollary 3.1.5 (Upper semi-continuity of dimension). Let f : X → Y be a morphism of irreducible algebraic varieties. For all x ∈ X, define e(x) := dimx f −1 (f (x)) := max{dim W | W a component of f −1 (f (x)) containing x}. Then e is upper semi-continuous, i.e., for all integers n {e ≥ n} = {x ∈ X| e(x) ≥ n} is closed. Proof. cf. [Mum88, I. 8. Coro. 3]. Corollary 3.1.6 (Upper semi-continuity of dimension). Let f : X → Y be a morphism algebraic varieties. Then the function e : X −→ N x 7→ dimx f −1 (f (x)). is upper semi-continuous. Proof. Let X1 , . . . , Xk be the irreducible components of X. Applying Corollary 3.1.5 to the restrictions fj = f |Xj of f fj : Xj → fj (Xj ), we obtain that the functions ej (x) := dimx fj−1 (fj (x)) are all upper semi-continuous. Moreover ej (x) = dimx fj−1 (fj (x)) = dimx Xj ∩ f −1 (f (x)) . This implies e(x) = dimx f −1 (f (x)) = max{dimx Xj ∩ f −1 (f (x)) | x ∈ Xj } = max{ej (x)| x ∈ Xj }. It follows that {e ≥ n} =

k [

{ej ≥ n},

j=1

which is closed since so are the sets {ej ≥ n} and hence the corollary is proved. Corollary 3.1.7. Let f : X → Y be a morphism algebraic varieties and let e(x) = dimx f −1 (f (x)). If e(x) = i for all x ∈ X, then dim X = i + dim f (X).

70

Chapter 3. Modality

Proof. By Corollary 3.1.4, it suffices to prove the following inequality dim X ≥ i + dim f (X). Let Z be an irreducible component of f (X) s.t. dim Z = dim f (X). Let X1 , . . . , Xk be the irreducible components of f −1 (Z). Let I collect i = 1, . . . , k such that f (Xi ) = Y . Note that I 6= ∅ since Z is irreducible. Put [ [ Z 0 := Z \ f (Xi ) and X 0 := Xi . i6∈I

i∈I

Then Z 0 contains an open dense subset of Z: [ Z 0 ⊃ Z \ f (Xi ) i6∈I

and therefore dim Z 0 = dim Z = dim f (X). Besides, since the restrictions fi = f |Xi : Xi → Z 0 i ∈ I are dominant morphisms of irreducible varieties, there are open dense subsets Ui of Z 0 such that dim fi−1 (y) = dim Xi − dim Z 0 , ∀y ∈ Ui , T by Theorem 3.1.1.TThen i∈I Ui 6= ∅ since Z 0 is irreducible. Taking an element x ∈ X s.t. f (x) ∈ i∈I Ui we obtain [ f −1 (f (x)) = f −1 (f (x)) ∩ Xi i∈I

=

[

fi−1 (fi (x))

i∈I

and hence i = dimx f −1 (f (x)) = max{dimx fi−1 (fi (x))} i∈I

= max{dim Xi − dim Z 0 } i∈I

= max{dim Xi } − dim Z 0 i∈I

≤ dim X − dim f (X).

Corollary 3.1.8. Let f : X → Y be a morphism algebraic varieties and let e : X −→ N x 7→ dimx f −1 (f (x)). Then max{dim e−1 (i) − i} ≥ dim f (X). i≥0

3.2 G-modality

71

Proof. Put d := dim X. Then f (X) =

d [

f (e−1 (i))

i=0

and therefore dim f (X) = max dim f (e−1 (i)). i≤d

Taking i s.t. dim f (X) = dim f (e−1 (i)) and applying Corollary 3.1.7 to the restriction fi = f |e−1 (i) : e−1 (i) → Y we get dim e−1 (i) − i ≥ dim f (e−1 (i)) = dim f (X) and hence max{dim e−1 (i) − i} ≥ dim f (X). i≥0

3.2

G-modality

The purpose of this section is to make the notion of modality precise in the case of hypersurfaces over an algebraically closed field K of arbitrary characteristic. To do this we introduce the “number of parameters” and the “modality” for the action of an algebraic group G on the algebraic variety X, and show that both notions coincide. As consequences we give nice properties of modality which are very useful for calculations later. The following definition makes Arnol’d’s modality, given in [AGV85], precise. Definition 3.2.1. Let U ⊂ X be an open neighbourhood of x ∈ X and W be constructible in X. We introduce dimx W := max{dim Z | Z an irreducible component of W containing x}, U (i) := UG (i) := {y ∈ U | dimy (U ∩ G · y) = i}, i ≥ 0, G-par(U ) := max{dim U (i) − i}, i≥0

and call G-par(x) := min{G-par(U ) | U a neighbourhood of x} the number of G-parameters of x. Now we use a Rosenlicht stratification of X under the action of G to give the definition of modality. By Rosenlicht [Ros56, Theorem 2] (see also [Ros63]) there exists an open dense subset X1 ⊂ X, which is invariant under G s.t. X1 /G is a geometric quotient (cf. [MFK82, §1]), in particular, the orbit space X1 /G is an algebraic variety and the projection p1 : X1 → X1 /G, x 7→ [x], is a surjective morphism. Since X \ X1 is a variety of lower dimension, which is invariant under G, we can apply the theorem of Rosenlicht to X \ X1 and get an invariant open dense

72

Chapter 3. Modality

subset X2 ⊂ X \ X1 s.t. X2 /G is a geometric quotient. Continuing in this way with X3 ⊂ (X \ X1 ) \ X2 , we can finally write X as finite disjoint union of G-invariant locally closed algebraic subvarieties Xi , i = 1, . . . , s, such that Xi /G is a geometric quotient with quotient morphism pi : Xi → Xi /G. We call {Xi , i = 1, . . . , s} a Rosenlicht stratification of X under G. Note that a Rosenlicht stratification is by no means unique and that the proof of Rosenlicht, which works for arbitrary G, is not constructive. Definition 3.2.2. Let {Xi , i = 1, . . . , s} be a Rosenlicht stratification of the algebraic variety X under the action of an algebraic group G with quotient morphisms pi : Xi → Xi /G, and let U be an open neighbourhood of x ∈ X. We define G-mod(U ) := max {dim pi (U ∩ Xi ) }, 1≤i≤s

and call G-mod(x) := min{G-mod(U ) | U a neighbourhood of x} the G-modality of x. Proposition 3.2.3. We have G-par(U ) = G-mod(U ) and therefore G-par(x) = G-mod(x) for all x ∈ X. This is a special case of Proposition 3.3.3, taking h the identity morphism. It shows in particular that G-mod(U ) and G-mod(x) are independent of the Rosenlicht stratification. Let G act on the variety X. The following nice properties of the modality (Prop. 3.2.4-3.2.7) are very useful for our calculations later. Proposition 3.2.4. If X is irreducible, then G-mod(x) ≥ dim V − dim G. Proof. Let U be an open neighbourhood of x in X such that G-mod(U ) = G-mod(x). By Proposition , G-mod(U ) = max{dim U (i) − i}. i≥0

We first show that G-mod(U ) = max{dim U (≤ i) − i}, i≥0

where U (≤ i) := {y ∈ U | dimy (U ∩ G · y) ≤ i}, i ≥ 0. In fact, the inequality G-mod(U ) ≤ max{dim U (≤ i) − i} i≥0

follows from the inclusion U (i) ⊂ U (≤ i). Take an i0 such that max{dim U (≤ i) − i} = dim U (≤ i0 ) − i0 . i≥0

3.2 G-modality

73

Then max{dim U (≤ i) − i} = dim U (≤ i0 ) − i0 i≥0

= max dim U (i) − i0 i≤i0

≤ max dim U (i) − i i≤i0

≤ G-mod(U ). Now we set i1 := dim G. Then G-mod(U ) = max{dim U (≤ i) − i} i≥0

≥ dim U (≤ i1 ) − i1 = dim U − i1 = dim X − dim G.

Proposition 3.2.5. Let the affine algebraic group G act on the variety X and let x, y ∈ X such that G · x = G · y. Then G-mod(x) = G-mod(y). Proof. Since x and y play the same role, it is sufficient to show that G-mod(x) ≥ G-mod(y). Let (Xi , pi ), i = 1, . . . , s be a Rosenlicht stratification of X under G and U a neighbourhood of x such that G-mod(x) = G-mod(U ). Since G · x = G · y, there exists h ∈ G such that h · x = y. It is easy to see that V := h · U is a neighbourhood of y in X. Hence G-mod(x) = G-mod(U ) = max{dim pi (U ∩ Xi )} i

≥ max{dim pi h · (U ∩ Xi ) } i = max{dim pi (h · U ) ∩ Xi } i

= G-mod(V ) ≥ G-mod(y). This completes the proposition. Proposition 3.2.6. If the subvariety X 0 ⊂ X is invariant under G and if x ∈ X 0 , then G-mod(x) in X ≥ G-mod(x) in X 0 . Equality holds if G-mod(x) in X 0 ≥ G-mod(y), ∀y ∈ X \ X 0 . Proof. Let (Xi , pi ), i = 1, . . . , s resp. (Xi , pi ), i = s+1, . . . , k be a Rosentlich stratification of X 0 resp. of X\X 0 under G. Then X1 , . . . , Xk is a Rosentlich stratification of X under G. Take an open neighbourhood U ⊂ X s.t. G-mod(U ) = G-mod(x) in X.

74

Chapter 3. Modality

Hence G-mod(x) in X = G-mod(U ) = max {dim pi (U ∩ Xi )} i=1,...,k

≥ =

max {dim pi (U ∩ Xi )}

i=1,...,s

max {dim pi (U ∩ X 0 ) ∩ Xi }

i=1,...,s

= G-mod(U ∩ X 0 ) ≥ G-mod(x) in X 0 . Assume now that G-mod(x) in X 0 ≥ G-mod(y) in X, ∀y ∈ X \ X 0 . We first see that G-mod(x) ≥ dim Xj /G for every j = s + 1, . . . , s. In fact, for each j = s + 1, . . . , s we take an irreducible component Zj of Xj /G such that dim Zj = dim Xj /G. Then Wj := p−1 j (Zj ) is a irreducible component Zj of Xj . Take an yj ∈ Wj and an open neighbourhood Uj of yj in X such that G-mod(yj ) in X = G-mod(Uj ). Then by assumption, G-mod(x) ≥ G-mod(yj ) since yj ∈ Wj ⊂ X \ X 0 = G-mod(Uj ) = max {dim pi (Uj ∩ Xi )} i=1,...,k

≥ dim pj (Uj ∩ Xj ) ≥ dim pj (Uj ∩ Wj ) = dim Zj = dim Xj /G. Next, take an open neighbourhood W ∩X 0 of x in X 0 such that G-mod(W ∩X 0 ) = G-mod(x) in X 0 . It follows that G-mod(x) in X ≤ G-mod(W ) = max {dim pi (W ∩ Xi )} i=1,...,k

= =

max {dim pi (W ∩ Xi )}

i=1,...,s

max {dim pi (W ∩ X 0 ) ∩ Xi }

i=1,...,s

= G-mod(W ∩ X 0 ) in X 0 = G-mod(x) in X 0 and hence G-mod(x) in X = G-mod(x) in X 0 .

Proposition 3.2.7. Let the algebraic group G0 act on the variety X 0 and let p : X → X 0 be a morphism of varieties.

3.2 G-modality

75

(i) If p is open and if G · x ⊂ p−1 G0 · p(x) , ∀x ∈ X. Then G-mod(x) ≥ G0 -mod(p(x)), ∀x ∈ X. (ii) If p is equivariant (i.e. G · x = p−1 G0 · p(x) , ∀x ∈ X), then G-mod(x) ≤ G0 -mod(p(x)), ∀x ∈ X. In particular, if p is open and equivariant then G-mod(x) = G0 -mod(p(x)), ∀x ∈ X. Proof. Let (Xi0 , p0i ), i = 1, . . . , s be a Rosentlich stratification of X 0 under G0 and let Xi = p−1 (Xi0 ). Then Xi is invariant under G since for every x ∈ Xi , p G · x ⊂ G0 · p(x) ⊂ Xi0 . Take a Rosenlicht stratification (Xi,j , pi,j ); j = 1, . . . , si of Xi under G. Then there exist morphisms p¯i,j such that the following diagram commutes Xi,j pi,j

/

Xi p¯i,j

Xi,j /G

p

/

/

Xi0

.

p0i

Xi0 /G0

(i) Let U be an open neighbourhood of x such that G-mod(x) = G-mod(U ). Then U 0 := p(U ) is an open neighbourhood of p(x) in X 0 since h is open. Let us show that si [ 0 0 0 pi U ∩ X i ⊂ p¯i,j pi,j U ∩ Xi,j . j=1

In fact, assume that x0 ∈ U 0 ∩ Xi0 . Then there exists x ∈ U such that p(x) = x0 ∈ Xi0 . It yields that x ∈ p−1 (Xi0 ) = Xi and therefore x ∈ U ∩ Xi,j for some us the claim. j. This gives S si 0 0 0 Since pi U ∩ Xi ⊂ j=1 p¯i,j pi,j U ∩ Xi,j , we have dim p0i

0

U ∩

Xi0

= max {dim p¯i,j pi,j U ∩ Xi,j

j=1,...,si

} ≤ max {dim pi,j U ∩ Xi,j }. j=1,...,si

Hence G0 -mod(p(x)) ≤ G0 -mod(U 0 ) =

max {p0i U 0 ∩ Xi0 }

1=1,...,s

≤ max{dim pi,j U ∩ Xi,j } i,j

= G-mod(U ) = G-mod(x).

76

Chapter 3. Modality

(ii) Since G · x = p−1 G0 · p(x) , ∀x ∈ X, it is easy to see that the morphisms p¯i,j are injective. Take a neighbourhood V 0 of p(x) in X 0 such that G0 -mod(p(x)) = G0 -mod(V 0 ) and set V := p−1 (V 0 ). It is easy to see that p0i

0

U ∩

Xi0

⊃

si [

p¯i,j pi,j U ∩ Xi,j .

j=1

This implies dim p0i V 0 ∩ Xi0 ≥ max {dim p¯i,j pi,j V ∩ Xi,j } = max {dim pi,j V ∩ Xi,j } j=1,...,si

j=1,...,si

since the p¯i,j are injective. Hence max {dim p0i V 0 ∩ Xi0 } i=1,...,s ≥ max{dim pi,j V ∩ Xi,j }

G0 -mod(p(x)) = G0 -mod(V 0 ) =

i,j

= G-mod(V ) ≥ G-mod(x).

The two following corollaries are proven by applying Proposition 3.2.7 for the identity morphism id : X → X 0 := X resp. the inclusion i : U ,→ X. Corollary 3.2.8. Let the affine algebraic groups G0 act on the variety X such that G · x ⊂ G0 · x for all x ∈ X. Then G-mod(x) ≥ G0 -mod(x) for all x ∈ X. Corollary 3.2.9. Let the U ⊂ X be a G-invariant open subset of X. Then G-mod(u) in U = G-mod(u) in X, ∀u ∈ U.

3.3

G-modality with respect to a morphism

In order to define right and contact modality of singularities w.r.t. unfolding, we generalize the previous notation to morphisms into a G-variety X. We study also relations between modality and modality w.r.t. morphisms. Definition 3.3.1. Let the algebraic group G act on the variety X, let h : X 0 → X be a morphism of algebraic varieties and let U 0 be an open neighbourhood of x0 ∈ X 0 . For u0 ∈ U 0 and each i ≥ 0 we define VG,h (u0 ) := {v 0 ∈ X 0 | G · h(v 0 ) = G · h(u0 )} = h−1 (G · h(u0 )), 0 (i) := {u0 ∈ U 0 | dimu0 U 0 ∩ VG,h (u0 ) = i}, U 0 (i) := UG,h 0 G-parh (U 0 ) := max{dim UG,h (i) − i}, i≥0

and call G-parh (x0 ) := min{G-parh (U 0 ) | U 0 a neighbourhood of x0 ∈ X 0 } the number of G-parameters of x0 w.r.t. h.

3.3 G-modality with respect to a morphism

77

Definition 3.3.2. Let {Xj , j = 1, . . . , s} be a Rosenlicht stratification of X under G with projections pj : Xj → Xj /G, let h : X 0 → X be a morphism of algebraic varieties and let U 0 be an open neighbourhood of x0 ∈ X 0 . Set Xj0 := h−1 (Xj ), Uj0 := U 0 ∩ Xj0 . We define G-modh (U 0 ) := max {dim pj (h(Uj0 )) }, j=1,...,s

and call G-modh (x0 ) := min{G-mod(U 0 ) | U 0 a neighbourhood of x0 in X 0 } the G-modality of x0 w.r.t. h. The following proposition shows that G-modh (U 0 ) and G-modh (x0 ) are independent of the Rosenlicht stratification. Proposition 3.3.3. We have G-parh (U 0 ) G-parh (x0 ) = G-modh (x0 ) for all x0 ∈ X 0 .

=

G-modh (U 0 ) and therefore

Proof. We consider the composition pj

h

hj : Uj0 −→ Xj −→ Xj /G. and note that 0 0 −1 0 0 0 0 0 h−1 j (hj (x )) = Uj ∩ h ((G · h(x ))) = Uj ∩ VG,h (x ), ∀x ∈ Uj . 0 By the upper semi-continuity of the functions ej : Uj0 → N, x0 7→ dimx0 h−1 j (hj (x )), −1 the sets ej (i) are locally closed and then

0

U (i) =

s [

e−1 j (i)

j=1

is constructible in X 0 for all i ≥ 0. Taking an i such that G-parh (U 0 ) = dim U 0 (i) − i and applying Corollary 3.1.7 we deduce −1 0 G-parh (U 0 ) = max{dim e−1 j (i) − i} = max{dim hj (ej (i)) } ≤ G-modh (U ). j

j

Let j ∈ {1, . . . , s} be such that G-modh (U 0 ) = dim pj (h(Uj0 )) = dim hj (Uj0 ). Then 0 0 G-parh (U 0 ) = max{dim U 0 (i)−i} ≥ max{dim(e−1 j (i))−i} ≥ dim hj (Xj ) = G-modh (U ), i

i

where the second inequality follows from Corollary 3.1.8. Hence G-parh (U 0 ) = G-modh (U 0 ).

78

Chapter 3. Modality Let the algebraic group G act on the variety X.

Corollary 3.3.4. Let h : X 0 → X be a morphism of algebraic varieties. Then G-modh (x0 ) ≤ G-mod(h(x0 )), ∀x0 ∈ X 0 . Equality holds if for every open neighbourhood U 0 of x0 , there exists an open neighbourhood U of h(x0 ) in X s.t. U ⊂ h(U 0 ). In particular, equality holds if h is open. Proof. Let {(Xi , pi ), i = 1, . . . , s} be a Rosenlicht stratification of X under G, let U be an open neighbourhood of h(x0 ) in X such that G-mod(h(x0 )) = G-mod(U ). Set U 0 := h−1 (U ), Xj0 := h−1 (Xj ) and Uj0 := U 0 ∩ Xj0 . Then G-modh (x0 ) ≤ G-modh (U 0 ) =

max {dim pj (h(Uj0 )) } i=1,...,s ≤ max {dim pj (U ∩ Xj ) } i=1,...,s

= G-mod(U ) = G-mod(h(x)), where the second inequality follows from the fact that h(U 0 ∩ Xj0 ) = h(U 0 ) ∩ Xj ⊂ U ∩ Xj . To prove the second statement it suffices to prove the inequality: G-modh (x0 ) ≥ G-mod(h(x0 )). Let V 0 be an open neighbourhood of x0 in X 0 such that G-modh (x0 ) = G-modh (V 0 ). Set Vj0 := V 0 ∩ Xj0 with Xj0 = h−1 (Xj ). By assumption, there exists an open neighbourhood V of h(x) in X such that V ⊂ h(V 0 ). Hence G-modh (x0 ) = G-modh (V 0 ) =

max {dim pj (h(Vj0 )) }

i=1,...,s

max {dim pj (h(V 0 ) ∩ Xj ) } i=1,...,s ≥ max {dim pj (V ∩ Xj ) } =

i=1,...,s

= G-mod(V ) ≥ G-mod(h(x)).

Corollary 3.3.5. Let h0

h

g : X 00 → X 0 → X be morphisms of algebraic varieties. Then G-modg (x00 ) ≤ G-modh (h0 (x00 )), ∀x00 ∈ X 00 . Equality holds if for every open neighbourhood U 00 of x00 , there exists an open neighbourhood U 0 of h0 (x00 ) in X 0 s.t. U 0 ⊂ h0 (U 00 ). In particular, equality holds if h0 is open.

3.3 G-modality with respect to a morphism

79

Proof. Let {(Xi , pi ), i = 1, . . . , s} be a Rosenlicht stratification of X under G and let Xj0 := h−1 (Xj ), Xj00 := g −1 (Xj ) = h0−1 (Xj0 ). Take an open neighbourhood U 0 of h0 (x00 ) in X 0 such that G-modh (h0 (x00 )) = G-modh (U 0 ) and put U 00 := h−1 (U 0 ). Then h0 (U 00 ∩ Xj00 ) = h0 (U 00 ) ∩ Xj0 ⊂ U 0 ∩ Xj0 . Hence G-modg (x00 ) ≤ G-modg (U 00 ) =

max {dim pj ◦ g(U 00 ∩ Xj00 ) }

i=1,...,s

max {dim pj ◦ h ◦ h0 (U 00 ∩ Xj00 ) } i=1,...,s ≤ max {dim pj ◦ h(U 0 ∩ Xj0 ) } =

i=1,...,s

= G-modh (U 0 ) = G-modh (h0 (x00 )), To prove the second statement it suffices to prove the inequality: G-modg (x00 ) ≥ G-modh (h0 (x00 )). Let V 00 be an open neighbourhood of x00 in X 00 such that G-modg (x00 ) = G-modg (V 00 ). By assumption, there exists an open neighbourhood V 0 of h0 (x00 ) in X 0 such that V 0 ⊂ h0 (V 00 ). Hence G-modg (x00 ) = G-modg (V 00 ) =

max {dim pj ◦ g(V 00 ∩ Xj00 )}

i=1,...,s

max {dim pj ◦ h ◦ h0 (V 00 ∩ Xj00 )} = max {dim pj ◦ h h0 (V 00 ) ∩ Xj0 ) } i=1,...,s ≥ max {dim pj ◦ h V 0 ∩ Xj0 ) } =

i=1,...,s

i=1,...,s

= G-modh (V 0 ) ≥ G-modh (h0 (x00 )).

Corollary 3.3.6. Let the algebraic groups G0 act on the varieties X 0 . Let h : Y → X and h0 : Y → X 0 be two morphisms of varieties such that h−1 (G · h(y)) ⊂ h0−1 (G0 · h0 (y)), ∀y ∈ Y. Then for any open subset V ⊂ Y we have G-modh (V ) ≥ G0 -modh0 (V ) and therefore G-modh (y) ≥ G0 -modh0 (y), ∀y ∈ Y. In particular, if h−1 (G · h(y)) = h0−1 (G0 · h0 (y)), ∀y ∈ Y , then G-modh (y) = G0 -modh0 (y), ∀y ∈ Y. Proof. We first show that G-modh (V ) = maxi≥0 {dim VG,h (≤ i) − i} with VG,h (≤ i) := {v ∈ V | dimv V ∩ VG,h (v) ≤ i}. In fact, the inequality G-modh (V ) ≤ maxi≥0 {dim VG,h (≤ i) − i} follows from Proposition 3.3.3 and the fact that VG,h (i) ⊂ VG,h (≤ i).

80

Chapter 3. Modality

Take an i0 such that max{dim VG,h (≤ i) − i = dim VG,h (≤ i0 ) − i0 . i≥0

Then max{dim VG,h (≤ i) − i = dim VG,h (≤ i0 ) − i0 i≥0

= max dim VG,h (i) − i0 i≤i0

≤ max dim VG,h (i) − i i≤i0

≤ G-parh (V ) = G-modh (V ) due to Proposition 3.3.3, and hence G-modh (V ) = maxi≥0 {dim VG,h (≤ i) − i}. By Proposition 3.3.3, G-modh0 (V ) = G-parh0 (V ) = max{dim VG0 ,h0 (i) − i}. i≥0

There exists an i1 such that G-modh0 (V ) = dim VG0 ,h0 (i1 ) − i1 . Note that VG0 ,h0 (i1 ) ⊂ VG,h (≤ i1 ) by the assumption h−1 (G · h(y)) ⊂ h0−1 (G0 · h0 (y)), ∀y ∈ Y. Then we have G-modh0 (V ) = dim VG0 ,h0 (i1 ) − i1 ≤ dim VG,h (≤ i1 ) − i1 ≤ G-modh (V ). since G-modh (V ) = maxi≥0 {dim VG,h (≤ i) − i}. Corollary 3.3.7. Let h : X 0 → X and hj : Yj → X, j = 1, . . . , k, be morphisms of varieties and let U 0 be open in X 0 satisfying, that for all x0 ∈ U 0 there exist an index j and y ∈ Yj such that G · h(x0 ) = G · hj (y). Then G-modh (U 0 ) ≤ max{G-modhj (Yj )} ≤ max dim Yj . j

j

Proof. Let {(Xi , pi ), i = 1, . . . , s} be a Rosenlicht stratification of X under G and let Xj0 := h−1 (Xj ), Yi,j := h−1 j (Xi ). Let us show that 0

pi ◦ h(U ∩

Xi0 )

⊂

k [ j=1

pi ◦ hj (Yi,j ), for all i = 1, . . . , s.

3.4 Right and contact modality

81

In fact, assume that x0 ∈ U 0 ∩ Xi0 . By assumption there exist an index j and y ∈ Yj such that G · h(x0 ) = G · hj (y). This implies that hj (y) ∈ G · hj (y) = G · h(x0 ) ⊂ Xi since h(x0 ) ∈ Xi and since Xi is G-invariant. It follows that y ∈ Yi,j and hence 0

pi ◦ h(U ∩

Xi0 )

⊂

k [

pi ◦ hj (Yi,j ).

j=1

Then we have G-modh (U 0 ) = max dim pi ◦ h(U 0 ∩ Xi0 ) i

≤ max dim pi ◦ hj (Yi,j ) i,j

= max max dim pi ◦ hj (Yi,j ) j

i

= max {G-modhj (Yj )} j

= max {dim Yj }. j

3.4

Right and contact modality

We apply the results of the previous section to define the notion of right resp. contact modality of a singularity f ∈ K[[x]] and the notion of right resp. contact modality of f w.r.t unfoldings. These numbers coincide if the considered unfoldings are complete (e.g. the semi-universal deformation with trivial section). Moreover we show that the modality is upper semi-continuous. In the last of this section we give a bound from below for the modality via the multiplicity and the number of variables. These results help us to bound and then compute the modality of a given singularity. The group R := AutK (K[[x]]) of automorphisms of the local algebra K[[x]] = K[[x1 , . . . , xn ]] is called the right group. The contact group K is the semi-direct product of AutK (K[[x]]) with the group (K[[x]])∗ of units in K[[x]]. These groups acts on K[[x]] by R × K[[x]] −→ K[[x]] (Φ, f ) 7→ Φ(f )

and

K × K[[x]] −→ K[[x]] ((Φ, u), f ) 7→ u · Φ(f ).

Two elements f, g ∈ K[[x]] are called right (∼r ) resp. contact (∼c ) equivalent iff they belong to the same R- resp. K-orbit. Note that neither R nor K are algebraic groups, as they are infinite dimensional. In order to be able to apply the results from the previous section, we have to pass to the jet spaces.

82

Chapter 3. Modality An element Φ in the right group R is uniquely determined by n power series ϕi := Φ(xi ) =

n X

aji xj + terms of higher oder

j=1

such that det(aji ) 6= 0. For each integer k we define the k-jet of Φ, Φk := (j k (ϕ1 ), . . . , j k (ϕn )), where j k (ϕi ) is the k-jet of ϕi , i.e. the image of ϕi in the k-th jet space Jk := m/mk+1 , m the maximal ideal of K[[x]], represented in K[[x]] by the power series expansion of ϕi up to order k. We call Rk := {Φk | Φ ∈ R} respectively Kk := {(Φk , j k u) | (Φ, u) ∈ K} the k-jet of R respectively of K. Note that Jk is an affine space and Rk , Kk are affine algebraic groups. These groups act algebraically on Jk by Kk × Jk −→ Jk (Φk , j k f ) 7→ j k (Φ(f ))

and

Kk × Jk −→ Jk ((Φk , j k u), j k f ) 7→ j k (u · Φ(f )).

Recall that for f ∈ K[[x]], let µ(f ) = dim K[[x]]/j(f ) denote the Milnor number and τ (f ) = dim K[[x]]/(hf i + j(f )) the Tjurina number of f , where j(f ) is the ideal in K[[x]] generated by all partials of f . Definition 3.4.1. Let f ∈ K[[x]] be such that µ(f ) < ∞ resp. τ (f ) < ∞ and let G = R resp. G = K. We define the G-modality of f , G-mod(f ), to be the Gk -modality of j k (f ) in Jk for sufficiently large integer k. We call f G-simple, G-unimodal, G-bimodal if G-mod(f ) equals to 0, 1, 2, respectively. Here, an integer k is sufficiently large for f w.r.t. G if there exists a neighbourhood U of j k (f ) in Jk s.t. every g ∈ K[[x]] with j k g ∈ U is k-determined w.r.t. G, which means that each h ∈ K[[x]] s.t. j k (h) = j k (g), is right resp. contact equivalent to g. Combining Propositions 3.4.6 and 3.4.14(ii) below we obtain that G-mod(f ) is independent of the sufficiently large k. The existence of a sufficiently large integer k for f w.r.t G will be shown in Proposition 3.4.4. Proposition 3.4.2. Let f ∈ K[[x]] be such that µ(f ) < ∞. Then K-mod(f ) ≤ R-mod(f ). Proof. Note that for any g ∈ K[[x]] we have R · g ⊂ K · g. The proposition follows from Corollary 3.2.8. We introduce some notions. Let T be an affine variety over K with thePstructure sheaf OT and its algebra of global section O(T ) = OT (T ). If F = aα x α ∈ O(T )[[x]], aα ∈ O(T ), then aα (t) ∈ K denotes the image of aα in OT,t /mt = K with mt the maximal ideal of the stalk OT,t , and therefore F (x, t) ∈ K[[x]] for each


83

t ∈ T . In the following we write ft (x) := F (x, t) instead of F , just to show the variables x and the parameter t ∈ T . Let f ∈ K[[x]] and t0 ∈ T . An element F (x, t) ∈ O(T )[[x]] is called an unfolding or deformation with trivial section of f at t0 ∈ T over T if F (x, t0 ) = f and ft ∈ m = hxi for all t ∈ T . The first statement of the following lemma says that the Milnor number µ and the Tjurina number τ are semi-continuous w.r.t. unfoldings. Its proof can be adapted from the construction in [GLS06, Thm. I.2.6] by applying [Har77, Thm. 12.8]. The second statement follows from (i). Lemma 3.4.3 (Semi-continuity of µ and τ ). (i) Let ft (x) = F (x, t) be an unfolding of f at t0 over an affine variety T . Then there exists an open neighbourhood U ⊂ T of t0 such that µ(ft ) ≤ µ(f ), resp. τ (ft ) ≤ τ (f ), ∀t ∈ U. (ii) The functions µ, τ are upper semi-continuous, i.e. for all i ∈ N, the sets Uµ,i := {f ∈ K[[x]] | µ(f ) ≤ i} resp. Uτ,i := {f ∈ K[[x]] | τ (f ) ≤ i} are open in K[[x]] w.r.t. the topology induced by the projections j k : K[[x]] → Jk . Proof. (ii) We shall show that the set V := {f ∈ K[[x]]| µ(f ) ≤ n} is open in K[[x]]. Take f ∈ V and a deformation X F (x, c) := fc (x) = f (x) + cα x α |α|≤2n

of f , where c = (cα )|α|≤2n ∈ J2n . By part (i), there exists a neighbourhood U2n of 0 in J2n such that µ(fc ) ≤ µ(f ) ≤ n, ∀c ∈ U2n . It follows from Corollary 1.1.11 that fc is right 2n-determined. That is, fc ∼r j 2n fc , ∀c ∈ U2n . Let U be the pre-image of j 2n f + U2n by the projections j 2n : K[[x]] → J2n . Then U is an open neighbourhood of f in K[[x]]. Besides, for every g ∈ U then j 2n g ∈ j 2n f + U2n , i.e. j 2n g = j 2n fc for some c ∈ U2n and therefore µ(j 2n g) = µ(j 2n fc ) = µ(fc ) ≤ n since fc ∼r j 2n fc . By again Corollary 1.1.11, j 2n g is right 2n-determined. This implies µ(g) = µ(j 2n g) ≤ n and hence g ⊂ V for all g ∈ U . This proves the proposition. Proposition 3.4.4. Let f ∈ K[[x]] be such that µ(f ) < ∞ (resp. τ (f ) < ∞) and let G = R (resp. G = K). Then all k ≥ 2 · µ(f ) (resp. k ≥ 2 · τ (f )) are sufficiently large for f w.r.t. R (resp. w.r.t. K). Proof. By the upper semi-continuity of µ, τ , the subsets Uµ := {g ∈ K[[x]] | µ(g) ≤ µ(f )} and Uτ := {g ∈ K[[x]] | τ (g) ≤ τ (f )} are open. It follows from Corollary 1.1.11 that g is k-determined w.r.t. G for all g ∈ Uµ (resp. Uτ ) and all k ≥ 2 · µ(f ) (resp. k ≥ 2 · τ (f )). This means that k is sufficiently large for f w.r.t. G.

84

Chapter 3. Modality

Definition 3.4.5. Let f ∈ K[[x]] be such that µ(f ) < ∞ (resp. τ (f ) < ∞). Let G = R (resp. G = K). Let ft (x) be an unfolding of f at t0 over an affine variety T . Let k be sufficiently large for f w.r.t. G and let Φk be the morphism of algebraic varieties defined by Φk : T → Jk , t 7→ j k ft (x). We define G-modF (f ) := Gk -modΦk (t0 ) and call it the G-modality or the number of G-parameters of f w.r.t. the unfolding F . Note that Gk acts on on Jk and that Gk -modΦk (t0 ) is understood in the sense of Definition 3.3.2. Proposition 3.4.6. The number G-modF (f ) is independent of the sufficient large integer k for f w.r.t. G. Proof. Let UG denotes the open neighbourhood Uµ resp. Uτ of f in K[[x]], defined as in the proof of Proposition 3.4.4. It is easy to see that the map Φ : T −→ K[[x]], t 7→ ft (x) is continuous. Then the pre-image VG = Φ−1 (UG ) is an open neighbourhood of t0 . For each k sufficient large for f w.r.t. G we consider the map i

Φ

jk

ϕk : VG ,→ T −→ K[[x]] −→ Jk . By Corollary 3.3.5, G-modΦk (t0 ) = G-modϕk (t0 ) since Φk = j k ◦ Φ. If k1 , k2 are both sufficient large for f w.r.t. G, then we can easily check that −1 ϕ−1 k1 Gk1 · ϕk1 (t) = ϕk2 Gk2 · ϕk2 (t) , ∀t ∈ VG . Corollary 3.3.6 yields that Gk1 -modϕk1 (t0 ) = Gk2 -modϕk2 (t0 ) and hence Gk1 -modΦk1 (t0 ) = Gk2 -modΦk2 (t0 ), which proves the proposition. Proposition 3.4.7 (Semicontinuity of modality). Let G = R (resp. G = K). Then the G-modality is upper semicontinunous, i.e. for all i ∈ N, the sets Ui := {f ∈ K[[x]]|G-mod(f ) ≤ i} is open in K[[x]]. Consequently, the G-modality is upper semicontinunous under unfoldings, i.e. for any unfolding ft (x) := F (x, t) at t0 over T of f with µ(f ) < ∞ (resp. τ (f ) < ∞) then the set Uf := {t ∈ T |G-mod(ft ) ≤ G-mod(f )} is open in T . Proof. Let f ∈ Ui , let k be sufficiently large for f w.r.t. G , and let U 0 be an open neighbourhood of j k f in Jk such that k is sufficiently large for all g ∈ K[[x]] with j k g ∈ U 0 . Then for every g ∈ U 0 , G-mod(g) = Gk -mod(j k g). Take an open neighbourhood U 00 of j k f in Jk such that Gk -mod(j k f ) = Gk -mod(U 00 ) = i and set U := (j k )−1 (U 0 ∩ U 00 ). It is easy to see that U is an open neighbourhood of f in K[[x]] and U ⊂ Ui . This implies that Ui is open in K[[x]].


85

So far we considered families of singularities parametrized by (affine) varieties, in particular by sufficiently height jet spaces. Now we want to use the semiuniversal deformation of a singularity since its base space has much smaller dimension. However for moduli problems, the formal deformation theory is not sufficient. We have to pass to the étale topology and apply Artin’s resp. Elkik’s algebraization theorems. Recall that an étale neighbourhood of a point x in a variety X consists of a variety U with a point u ∈ U and an étale morphism ϕ : U → X with ϕ(u) = x (see, [Mum88, Definition III.5.1]). ϕ is a morphism of pointed varieties, usually denoted by ϕ : U, u → X, x. The connected étale neighbourhoods form a filtered system and the direct limit ˜X,x := lim OU,u = lim O(U ) O −→

−→

is called the Henselization (see [Na53], [Ra70], [KPPRM78]) of OX,x . We have ˆX,x = O Û,u where ∧ denotes the completion w.r.t. the maximal ideal. The O Henselization of K[x]hxi is the ring Khxi of algebraic power series in x = (x1 , . . . , xn ). K[x] ⊂ Khxi ⊂ K[[x]] and Khxi may be considered as the union of O(U ) ⊂ K[[x]] or OU,u ⊂ K[[x]]. More precisely Remark 3.4.8. For each finite subset A ⊂ Khxi there exists an étale neighbourhood ϕ : U, u0 → An , 0 such that A ,→ O(U ), i.e. for all algebraic power series a(x) ∈ A we have a(ϕ(u)) ∈ O(U ). Lemma 3.4.9. Let x = (x1 , . . . , xn ) and x = (y1 , . . . , ym ). Let Khxi resp. Khx, yi the henselization of K[x]hxi resp. K[x, y]hx,yi . Then Khx, yi ⊂ Khxi[[y]]. Proof. It suffices to show that Khx, y1 i ⊂ Khxi[[y1 ]] or, equivalently, if X f= ai (x)y1i ∈ Khx, y1 i i≥0

then ai (x) ∈ Khxi for all i ≥ 0. We argue by induction on i. If i = 0 then a0 (x) = f (x, 0). Since f ∈ Khx, y1 i, there exists a polynomial p(t) =

k X

bj (x, y1 )tj ∈ (K[x, y1 ][t]) with bk = 1

j=1

such that p(f ) = 0. This implies that k X

j

bj (x, 0)(f (x, 0)) =

j=1

k X

bj (x, 0)(aα (x))j = 0,

j=1

and hence a0 (x) ∈ Khxi. We now prove the induction step. Suppose that aν (x) ∈ Khxi for all ν < i. It yields that X f (i) (x, y1 ) := al (x)y1 l ∈ Khx, y1 i, l≥i

86

Chapter 3. Modality

i.e. there exists q(t) =

k X

bj (x, y1 )tj ∈ K[x, y1 ][t] with bk = 1

j=1

such that q(f (i) ) = 0. We write m

with mj = ordy1

mj +1

bj (x, y1 ) := bj,mj (x)y1 j + bj,mj +1 (x)y1 bj (x, y1 ) , and define

+ ...,

m0 := min {mj + j · i}; J := {j|mj + j · i = m0 }. j=0,...,k

Since q(f (i) ) = 0, X

bj,mj (x)(ai (x))j = 0,

j∈J

i.e. ai (x) is a root of the polynomial proof.

P

j∈J

bj,mj (x)tj ∈ K[x][t]. This completes the

Definition 3.4.10. Let F (x, t) be an unfolding of f at t0 over an affine variety T . (a) An unfolding H(x, s) at s0 over an affine variety S of f is called a pullback or an induced unfolding of F if there exists a morphism ϕ : S, s0 → T, t0 s.t. H(x, s) = F (x, ϕ(s)). (b) An unfolding H(x, s) at s0 over an affine variety S of f is called a (étale) G-pullback or a (étale) G-induced unfolding of F if there exist an étale neighbourhood ϕ : W, w0 → S, s0 and a morphism ψ : W, w0 → T, t0 such that G · H(x, ϕ(w)) = G · F (x, ψ(w)) for all w ∈ W . (c) The unfolding F (x, t) is called G-complete if any unfolding of f is a G-pullback of F . The following lemma is an immediate consequence of the definition. Lemma 3.4.11. Let H be a G-pullback of the unfolding F . If H is G-complete, then so is F . Proposition 3.4.12. Let ft1 (x) and ft2 (x) be unfoldings of f 1 , f 2 at t0 ∈ T such that j k ft1 (x) = j k ft2 (x) for all t ∈ T , where k is sufficiently large for both f 1 and f 2 w.r.t. G. Then ft1 is G-complete if and only if so is ft2 . Proof. It suffices to prove that if ft1 is G-complete then so is ft2 . Let h2s be any unfolding of f 2 at s0 ∈ S. Then h1s (x) := h2s (x) + f 1 (x) − f 2 (x) is an unfolding of f 1 at s0 ∈ S. Since ft1 is G-complete, there exist an étale neighbourhood ϕ1 : W 1 , w0 → S, s0 and a morphism ψ 1 : W 1 , w0 → T, t0 such that


87

G · h1ϕ1 (w) (x) = G · fψ11 (w) (x) for all w ∈ W 1 . On the other hand, since k is sufficiently large for both f 1 and f 2 w.r.t. G, there exists an open neighbourhood U ⊂ T of t0 such that ft1 and ft2 are both k-determined w.r.t. G for all t ∈ U . We set W 2 := W 1 ∩ (ψ 1 )−1 (U ), ϕ2 := ϕ1 |W 2 and ψ 2 := ψ 1 |W 2 . Then G · h2ϕ2 (w) (x) = G · h1ϕ2 (w) (x) and G · fψ12 (w) (x) = G · fψ22 (w) (x) since j k ft1 = j k ft2 and j k h1s = j k h2s . Hence G · h2ϕ2 (w) (x) = G · h1ϕ2 (w) (x) = G · fψ12 (w) (x) = G · fψ22 (w) (x). This implies that ft2 is G-complete. Proposition 3.4.13. Any singularity f with µ(f ) < ∞ (resp. τ (f ) < ∞) has a right (resp. contact) complete unfolding. More precisely, if k is sufficiently large for f w.r.t. RP (resp. w.r.t. K), then the unfolding of f over Jk = AN (with the α identification: |α|≤k cα x 7→ (cα )|α|≤k ), fc (x) := F (x, c) = f (x) +

X

cα x α ,

|α|≤k

is right (resp. contact) complete. Proof. Since k is sufficiently large for f w.r.t. right (resp. contact) equivalence, there exists a neighbourhood U ⊂ Jk of j k f such that each g ∈ U is right (resp. contact) k-determined. Let hs (x) := H(x, s) be an arbitrary unfolding of f at s0 over S and let W := ψ −1 (U ) be the pre-image of U by the morphism ψ : S −→ AN , s 7→ j k hs (x) − j k f (x). Then H(x, s) is right (resp. contact) equivalent to F (x, ψ(s)) for all s ∈ W since j k H(x, s) = j k F (x, ψ(s)) ∈ U and hence H is a right (resp. a contact) pullback of F , which proves the proposition. Proposition 3.4.14. Let f ∈ K[[x]] be such that µ(f ) < ∞ (resp. τ (f ) < ∞) and let G = R (resp. G = K). Let ft (x) := F (x, t) be an unfolding of f at t0 over T . (i) If the unfolding hs (x) := H(x, s) of f at s0 over S, is a G-pullback of F , then G-modF (f ) ≥ G-modH (f ). (ii) We always have G-mod(f ) ≥ G-modF (f ). Equality holds if F (x, t) is Gcomplete. Proof. (i) Since H is a G-pullback of F , there exist an étale neighbourhood ϕ : W, w0 → S, s0 and a morphism ψ : W, w0 → T, t0 such that G · H(x, ϕ(w)) = G · F (x, ψ(w)) for all w ∈ W .

88

Chapter 3. Modality Let k be sufficiently large for f and let Φk and Ψk be the morphisms defined by Φk : T → Jk , t 7→ j k ft (x) and Ψk : S → Jk , s 7→ j k hs (x).

Then Φ−1 (G·Φ(w)) = Ψ−1 (G·Ψ(w)) for all w ∈ W with Φ := Φk ◦ψ and Ψ := Ψk ◦ϕ. Corollary 3.3.6 yields that Gk -modΦ (w0 ) = Gk -modΨ (w0 ). Applying Corollary 3.3.5 to compositions Φ = Φk ◦ ψ and Ψ = Ψk ◦ ϕ we get Gk -modΦk (t0 ) ≥ Gk -modΦ (w0 ) and Gk -modΨ (w0 ) = Gk -modΨk (s0 ). Hence G-modF (f ) = Gk -modΦk (t0 ) ≥ Gk -modΨk (s0 ) = G-modH (f ). (ii) Since k is sufficiently large for f w.r.t. G, G-mod(f ) = Gk -mod(j k f ) and G-modF (f ) = Gk -modΦk (t0 ). It follows from Corollary 3.3.4 that Gk -mod(j k f ) ≥ Gk -modΦk (t0 ). Hence G-mod(f ) ≥ G-modF (f ). P α via the map |α|≤k cα x 7→ (cα )|α|≤k and consider the

We identify Jk with K N unfolding X hc (x) := H(x, c) = f (x) + cα x α |α|≤k N

of f at 0 over K . Since the translation ψ : K N −→ Jk = KN c 7→ j k hc (x) = j k f + c is an isomorphism, it follows from Corollary 3.3.4 that Gk -mod(j k f ) = Gk -modψ (0) and hence G-mod(f ) = G-modH (f ). Now, if the unfolding F is G-complete, then H is a G-pullback of F . By (i), G-modH (f ) ≤ G-modF (f ) and hence G-mod(f ) = G-modF (f ). Example 3.4.15. (a) The unfolding ft (x) := F (x, t) = xp+1 + t1 x + . . . + tp xp of f = xp+1 over T = Ap , is right complete. Indeed, for any unfolding ht (x) = H(x, s) of f at s0 over S we write H(x, s) = a1 (s)x + . . . + ap (s)xp + ap+1 (s)xp+1 + . . . with ai (s) ∈ O(S). Then ai (s0 ) = 0 ∀i ≤ p and ap+1 (s0 ) = 1. Consider the open neighbourhood W := {s ∈ S | ap+1 (s) 6= 0} of s0 in S and the morphism


89

ϕ : W → T, s 7→ a0 (s)/ap+1 (s), . . . , ap (s)/ap+1 (s) . Then j p+1 fϕ(s) (x) = j p+1 hs (x) for all s ∈ W . It follows from Corollary 1.1.11 that F (x, ϕ(s)) ∼r H(x, s), for each s ∈ W. Note that {x, . . . , xp } is a basis of m/mj(f ) and that F is a semiuniversal deformation of f with trivial section by Proposition 3.4.16. (b) The right semi-universal deformation of f = xp+1 ∈ K[[x]] with char(K) = p > 2 is given by H(x, t) = xp+1 + t1 x + . . . + tp−1 xp−1 . This unfolding of f over Ap−1 is not right complete. In fact, it is not difficult to see that H(x, t) is equivalent to one of {x, . . . , xp−1 , xp+1 } for t ∈ Ap . Corollary 3.4.18 yields that R-modH (f ) = 0, while R-mod(f ) = 1 by Theorem 4.2.8 and hence H is not right complete due to Proposition 3.4.14. To see this directly, consider the family xp+1 + sxp in characteristic p > 0 over K which, as an unfolding with trivial section, cannot be induced by a morphism ϕ : A1 → Ap−1 : Since H(x, ϕ(s)) has multiplicity 6= p in K[[x]] for all s 6= 0, it cannot be right equivalent to xp+1 + sxp which has multiplicity p for s 6= 0. This is of course not a contradiction to F being versal as deformation without section. This means that the family xp+1 + sxp ∈ K[[x, s]] can be induced from H by a morphism ϕ : K[[t1 , . . . , tp−1 ]] → K[[s]] (up to right equivalence in K[[x, s]] over K[[s]]). In fact, define ϕ by t1 7→ −sp , ti 7→ 0 for i > 1, then, if char(K) = p, H(x, ϕ(s)) = −sp x + xp+1 = (x − s)p x ∼r xp+1 + sxp , via the isomorphism Φ : K[[x, s]] → K[[x, s]] over K[[s]], given by x 7→ x − s, s 7→ s. However, Φ does not respect the trivial section. s If char(K) 6= p, we can use the Tschirnhaus transformation x 7→ x − to p eliminate sxp from xp+1 + sxp and to show that xp+1 + sxp can be induced from H. The following proposition is stronger than Proposition 3.4.13 because it reduces the number of parameters of a G-complete unfolding considerably. For the proof we need the nested Artin approximation theorem (cf. [Art68], [Art69], [Po86]). Proposition 3.4.16. Let f ∈ m ⊂ K[[x]] be such that µ(f ) < ∞ (resp. τ (f ) < ∞). Let g1 , . . . , gl ∈ K[x] be a basis of the algebra m/m · j(f ) (resp. of m/hf i + m · j(f )). Then the unfolding (with trivial section) of f over Al , F (x, t) = f (x) +

l X

ti gi (x),

i=1

represents a formally semiuniversal deformation of f with trivial section with respect to right (resp. contact) equivalence. Moreover, it is right (resp. contact) complete. Proof. We first show that F represents a formally semiuniversal deformation of f with trivial section with respect to right (resp. contact) equivalence. Indeed in [BGM12, Proposition 2.7] it is shown that the tangent space to the base space of the semiuniversal deformation with trivial section is m/m · j(f ) (resp. m/hf i + m · j(f )). The proof of the existence of a semiuniversal deformation in [KaS72] or [GLS06,

90

Chapter 3. Modality

Thm. II.1.16] can be easily adapted to deformations with section, showing the versality of F and hence proving the first claim. Let G = R (resp. G = K) and let k ≥ 2µ(f ) (resp. k ≥ 2τ (f )) be sufficiently large for f w.r.t. G. In the proof of the following claims we will restrict ourselves to the case of contact equivalence, since the case of right equivalence can be treated analogously. Claim 1: If j k g = j k f then m · j(g) = m · j(f ) (resp. m/hgi + m · j(g) = m/hf i + m · j(f )). Proof: It suffices to prove that m/hgi + m · j(g) ⊂ m/hf i + m · j(f ). In fact, since k is sufficiently large for f w.r.t. contact equivalence, f is contact k-determined and then mk+1 ⊂ hf i + m · j(f ) by Theorem 1.1.10. It follows that g = (g − f ) + f ∈ hf i + m · j(f ) and xi ·

∂(g − f ) ∂f ∂g = xi · + xi · ∈ hf i + m · j(f ), ∀i, j ∂xj ∂xj ∂xj

since (g − f ) ∈ mk+1 . This proves the claim. Claim 2: The system {j k g1 , . . . , j k gl } is a basis of the algebra m/m · j(j k f ) (resp. of m/hj k f i + m · j(j k f )). Proof: We have m/hj k f i + m · j(j k f ) = m/hf i + m · j(f ) by Claim 1. Moreover, gj = j k gj

mod m/hf i + m · j(f )

since mk+1 ⊂ hf i+m·j(f ). Hence {j k g1 , . . . , j k gl } is a basis of the algebra m/hj k f i+ m · j(j k f ). Combining Claim 1, Claim 2 and Proposition 3.4.12 we may replace F resp. f by j k F resp. j k f and assume that F ∈ K[x, t] and f ∈ K[x]. Consider the complete unfolding of f over AN = Jk X hc (x) := H(x, c) = f (x) + cα x α |α|≤k

as in Proposition 3.4.13. By Lemma 3.4.11 the proof is completed by showing that H is a G-pullback of F . ˆ = In fact, the versality of F implies that there exist formal power series Φ n l ˆ 1, . . . , Φ ˆ n ) ∈ hxiK[[c, x]] , ϕˆ = (ϕˆ1 , . . . , ϕˆl ) ∈ hciK[[c]] and a unit uˆ(c, x) ∈ (Φ K[[c, x]] (with uˆ = 1 for G = R) with uˆ(0, 0) 6= 0 and

det

ˆ i ∂Φ |(0,0) 6= 0, ∂xj

(3.4.1)

such that ˆ x , c) = uˆ(c, x) · F x, ϕ(c) H Φ(c, ˆ . Let y = (y1 , . . . , yn+l+1 ) be new indeterminates (omitting yn+l+1 if G = R) and let P (x, c, y) = H(y1 , . . . , yn , c) − yn+l+1 · F (x, yn+1 , . . . , yn+l ) ∈ K[x, c, y].


91

ˆ (c, x) ∈ K[[c, x]] The formal versality of F implies that P = 0 has a formal solution y with yî := yˆn+j := yˆn+l+1 :=

ˆ i ∈ Khc, xi, 1 ≤ i ≤ n, Φ ϕˆj ∈ Khci, 1 ≤ j ≤ l, uˆ ∈ Khc, xi.

By the nested Artin Approximation Theorem ([Po86, Theorem 1.4]), there exists a ˜ (c, x) ∈ Khc, xi of P = 0, i.e. P (x, c, y ˜ (x, c)) = 0 such that solution y y˜i ∈ Khc, xi, 1 ≤ i ≤ n, yñ+j ∈ Khci, 1 ≤ j ≤ l, yñ+l+1 ∈ Khc, xi and ˜ (c, x) − y ˆ (c, x) ∈ hc, xi2 . y

(3.4.2)

Passing to the k-jet spaces by projection j k : K[[x, c]] → K[[x, c]]/hxik+1 we get ˜ ) = j k P (x, c, j k y˜1 , . . . , j k yñ+l+1 ) = 0. j k P (x, c, y Let A be the set of the coefficients of xα , |α| ≤ k which appear in all j k (˜ yi )(x, c), i = 1, . . . , n + l + 1. Then A ⊂ Khci by Lemma 3.4.9. It follows from Remark 3.4.8 that there exists an étale neighbourhood ϕ : U, u0 → AN , 0 such that a(ϕ(u)) ∈ O(U ) for all a(c) ∈ A. This implies that j k P (x, ϕ(u), j k (˜ y(ϕ(u), x))) = 0 in O(U )[[x]]/hxik+1 . Combining (2.1), (2.2) we obtain that there exists an open neighbourhood W 0 ⊂ U of u0 such that for all u ∈ W 0 , Φ(u, x) := j k y˜1 (ϕ(u), x), . . . , j k yñ (ϕ(u), x) ∈ Aut(K[[x]]), ψ(u) := yñ+1 (ϕ(u), . . . , yñ+l (ϕ(u) ∈ O(U )l and u(u, x) := j k yñ+l+1 (ϕ(u), x) ∈ K[[x]]∗ , i.e. G · j k H(Φ(u, x), ϕ(u)) = G · F (x, ψ(u)). Moreover since k is sufficiently large for f w.r.t. G, it is not difficult to see that there exists an open affine neighbourhood W ⊂ W 0 of u0 such that G · H(Φ(u, x), ϕ(u)) = G · j k H(Φ(u, x), ϕ(u)), ∀u ∈ W. Hence G · H(Φ(u, x), ϕ(u)) = G · j k H(Φ(u, x), ϕ(u)) = G · F (x, ψ(u)), ∀u ∈ W, which proves the claim.

92

Chapter 3. Modality

Proposition 3.4.17. Let f ∈ K[[x]] be such that µ(f ) < ∞ (resp. τ (f ) < ∞) and let G = R (resp. G = K). Let ft (x) := F (x, t) be any unfolding of f at t0 over an affine variety T . Let T (i) , i = 1, . . . , q be affine varieties and let H (i) (x, t(i) ) ∈ O[T (i) ][[x]]. Assume that there exists an open neighbourhood t0 ∈ V ⊂ T satisfying: for every t ∈ V there exist an i = 1, . . . , q and a t(i) ∈ T (i) such that F (x, t) ∼r H (i) (x, t(i) ). Then G-modF (f ) ≤ max Gk -modH (i) (T (i) ) ≤ max {dim T (i) }. i=1,...,q

i=1,...,q

Proof. Let k be sufficiently large for f w.r.t. G. Considering the morphisms (i)

Φk : V → Jk , t 7→ j k ft (x) and Φk : T (i) → Jk , s 7→ j k h(i) s (x), i = 1, . . . , q, and applying Corollary 3.3.7 we obtain Gk -modΦk (V ) ≤ max {Gk -modΦ(i) (T (i) )} ≤ max {dim T (i) }. i=1,...,q

k

i=1,...,q

Hence G-mod(f ) = Gk -modΦk (t0 ) ≤ Gk -modΦk (V ) ≤ max {dim T (i) }. i=1,...,q

Combining Propositions 3.4.14 and 3.4.17 we get Corollary 3.4.18. Let f ∈ K[[x]] be such that µ(f ) < ∞ resp. τ (f ) < ∞. f is Gsimple iff it is of finite G-unfolding type, i.e. there exists a finite set F of G-classes of singularities satisfying: for any (or, equivalently, for one G-complete) unfolding F (x, t) of f at t0 over an affine variety T , there exists a Zariski open neighbourhood V of t0 ∈ T , such that the set of G-classes of singularities of F (x, t), t ∈ V , belongs to the set F. Proposition 3.4.19. Let f ∈ K[[x]] = K[[x1 , . . . , xn ]] be such that µ(f ) < ∞ (resp. τ (f ) < ∞) and let G = R (resp. G = K). Let m = mt(f ) be the multiplicity of f . Then n+m−1 G-mod(f ) ≥ − n2 . m In particular, (i) if n = 2, then G-mod(f ) ≥ m − 3; m2 + 3m − 16 . (ii) if n ≥ 3 and m ≥ 3, then G-mod(f ) ≥ 2 Proof. Let k be sufficiently large for f w.r.t. G. Then k ≥ m and G-mod(f ) = Gk -mod(j k f ). Put X := mm /mk+1 ⊂ Jk . It follows from Proposition 3.2.6 that G-mod(f ) in Jk ≥ G-mod(f ) in X.


93

Let the linear group G0 := GL(n, K) act on X 0 := mm /mm+1 by G0 × X 0 → X 0 , (A, g(x)) 7→ g(Ax). Consider the projection p : X → X 0 . It is easy to see that p is open and G · g ⊂ p−1 G0 · p(g) for all g ∈ X. Then Proposition 3.2.7 yields G-mod(g) ≥ G0 -mod(p(g)), ∀g ∈ X. In order to prove the proposition, it is sufficiently to show that n+m−1 0 G -mod(p(g)) ≥ − n2 , ∀g ∈ X. m Indeed, it is easy to see that n+m−1 0 dim X = and dim GL(n, K) = n2 . m Hence GL(n, K)-mod(p(g)) ≥ dim X 0 − dim GL(n, K) n+m−1 = − n2 m by Proposition 3.2.4. This completes the proof. (i) and (ii) follow from explicit calculations.

Chapter 4 Right Classification of Singularities We classify right simple hypersurface singularities f ∈ m2 ⊂ K[[x1 , . . . , xn ]], K an algebraically closed field of characteristic p > 0. Arnol’d in [Arn72] classified right simple singularities for K = R and C. He showed that the simple singularities are exactly the ADE-singularities, i.e. the two infinite series Ak , k ≥ 1, Dk , k ≥ 4, and the three exceptional singularities E6 , E7 , E8 . It turned out later that the ADE-singularities of Arnol’d are also exactly those of modality 0 for contact equivalence. In the late eighties, Greuel and Kröning showed in [GK90] that the contact simple singularities over a field of positive characteristic are again exactly the ADE-singularities but with a few more normal forms in small characteristic. A classification w.r.t. right equivalence in positive characteristic however, was never considered so far. A surprising fact of our classification is that for any fixed p > 0 there exist only finitely many right simple singularities. For example, if p = 2 and n is even, there is just one right simple hypersurface, x1 x2 + x3 x4 + . . . + xn−1 xn , while for n odd no right simple singularity exist. A table with normal forms for any n ≥ 1 and any p > 0 is given in section 3 (Theorems 4.3.1 - 4.3.3). In section 4.2 we classify completely univariate power series w.r.t. right equivalence by explicit normal forms (Theorem 4.2.7). Moreover we show that the right modality of f is equal to the integer part of µ/p (Theorem 4.2.8), where µ is the Milnor number of f . As a consequence we prove in this case that the modality is equal to the inner modality, which is the dimension of the µ-constant stratum in an algebraic representative of the semiuniversal deformation with trivial section (Corollary 4.2.10).

4.1

Contact simple hypersurface singularities

This section is to recall the classification of contact simple singularities in positive characteristic in [GK90].

95

96

Chapter 4. Right Classification of Singularities

Proposition 4.1.1 (Greuel and Kröning). A hypersurface singularity f ∈ m2 ⊂ K[[x]] = K[[x1 , . . . , xn ]] is contact simple if and only if it is contact equivalent to one of the following forms (I) char(K) 6= 2 I.1. n = 2 Name Ak Dk E6 E06 E16 E7 E07 E17 E8 E08 E18 E28 E18

Normal form w.r.t. contact equivalence x2 + y k+1 k≥1 2 k−1 x y+y k≥4 3 4 x +y x3 + y 4 + x2 y 2 additionally in char = 3 3 3 x + xy x3 + xy 3 + x2 y 2 additionally in char = 3 x3 + y 5 x3 + y 5 + x2 y 3 additionally in char = 3 x3 + y 5 + x2 y 2 additionally in char = 3 3 5 4 x + y + xy additionally in char = 5

I.2. n ≥ 3 Normal form g(x1 , x2 ) + x23 + . . . + x2n

g is one of the singularities in list I.1.

(II) char(K) = 2. II.1. n = 2 Name A2m−1 A2m D2m D2m+1 E6 E7 E8

II.2. n = 3

A02m Ar2m D02m+1 Dr2m+1 E06 E16

Normal form x2 + xy m x2 + y 2m+1 x2 + y 2m+1 + xy 2m−r x2 y + xy m x2 y + y 2m x2 y + y 2m + xy 2m−r x3 + y 4 x3 + y 4 + xy 3 x3 + xy 3 x3 + y 5

m≥1 m ≥ 1, 1 ≤ r ≤ m − 1 m≥1 m≥2 m≥2 m ≥ 2, 1 ≤ r ≤ m − 1

4.2 Right classification of univariate power series Name Ak D2m D2m+1 E6 E7

E8

D02m Dr2m D02m+1 Dr2m+1 E06 E16 E07 E17 E27 E37 E08 E18 E28 E38 E48

Normal form z k+1 + xy z 2 + x2 y + xy m z 2 + x2 y + xy m + xy m−r z z 2 + x2 y + y m z z 2 + x2 y + y m z + xy m−r z z 2 + x3 + y 2 z z 2 + x3 + y 2 z + xyz z 2 + x3 + xy 3 z 2 + x3 + xy 3 + x2 yz z 2 + x3 + xy 3 + y 3 z z 2 + x3 + xy 3 + xyz z 2 + x3 + y 5 z 2 + x3 + y 5 + xy 3 z z 2 + x3 + y 5 + xy 2 z z 2 + x3 + y 5 + y 3 z z 2 + x3 + y 5 + xyz

97

m≥2 m ≥ 2, 1 ≤ r ≤ m − 1 m≥2 m ≥ 2, 1 ≤ r ≤ m − 1

II.3. n ≥ 4 Normal form g(x1 , x2 ) + x3 x4 + . . . + xn−1 xn if n is even g(x1 , x2 , x3 ) + x4 x5 + . . . + xn−1 xn if n is odd,

where g ∈ K[[x1 , x2 ]] resp. K[[x1 , x2 , x3 ]] is one of the list II.1 resp. II.2. Proof. It follows from Corollary 3.4.18 and [GK90, Def. 1.2, Thm. 1.4].

4.2 4.2.1

Right classification of univariate power series Normal forms of univariate power series

P n Let f = n≥0 cn x ∈ K[[x]] be a univariate power series, let supp(f ) := {n ≥ 0 | cn 6= 0} be the support of f and mt(f ) := min{n | n ∈ supp(f )} the multiplicity of f . If char(K) = 0 and if ϕ(x) = a1 x + a2 x2 + . . . , a1 6= 0, is a coordinate change, then the coefficients ai of ϕ can be determined inductively from the equation f (x) = c0 + (ϕ(x))mt(g) with g(x) := f − c0 . Hence f is right equivalent to c0 + xmt(g) . In the following we investigate f ∈ K[[x]] with char(K) = p > 0. The aim of this section is to give a normal form of f . It turns out that it depends in a complicated way from the divisibility relation between p and the support of f . To describe this relation we make the following definition, where later on ∆ will be supp(f ). For each subset ∆ ⊂ N \ {0}, we define (a) m(∆) := min{n|n ∈ ∆}

98


(b) e(∆) := min{e(n) | n ∈ ∆}, where e(n) := max{i|pi divides n}. (c) q(∆) := min{n | e(n) = e(∆), n ∈ ∆}. (d) k(∆) := 1 if m(∆) = q(∆), otherwise, k(∆) := max{k∆ (n) | m(∆) ≤ n < q(∆), n ∈ ∆}, where k∆ (n) := d

q(∆) − n q(∆) − n e denotes the ceiling of e(n) . e(∆) −p p − pe(∆)

pe(n)

¯ (e) d(∆) := 2q(∆) − m(∆) and d(∆) := q(∆) + pe(∆) k(∆) − 1 . ¯ ¯ (f) Λ(∆) := {n|m(∆) < n ≤ d(∆), e(∆) < e(n)} ∪ {q(∆)}. (g) If e(m(∆)) > e(∆) we define ∆0 := {n ∈ ∆| n < q(∆)}, e0 := e(∆0 ) and q0 := q(∆0 ). ¯ 0 ) ∩ N 0 and if e(m(∆)) = e0 , then Λ(∆) := {n ∈ N | q(∆) ≤ n ≤ d(∆), e(n) < e(m(∆))}. (Note that in this case one has k(∆) = k∆ (m(∆))). Case 2: If e(m(∆)) > e0 > 1 then Λ(∆) := Λ0 (∆) ∪ Λ1 (∆). Case 3: If e(m(∆)) > e0 = 1 and if k(∆) > k(q0 ), then Λ(∆) := Λ0 (∆)∪Λ1 (∆). 0 Case 4: If e(m(∆)) > e0 = 1 and k(∆) = k∆ (q0 ) and if k(∆0 ) ≥ b q−q c then p Λ(∆) := Λ0 (∆) ∪ Λ01 (∆). 0 c then Case 5: If e(m(∆)) > e0 = 1 and k(∆) = k∆ (q0 ) and if k(∆0 ) < b q−q p

Λ(∆) := Λ0 (∆) ∪ Λ001 (∆). Remark 4.2.1. The following facts (a)-(d) are immediate consequences of the definition. Property (e) follows from elementary calculations. (a) If p does not divide m(∆), then 1. e(∆) = e(m(∆)) = 0 and q(∆) = m(∆). 2. k(∆) = 0 and d(∆) = m(∆). (b) If e(m(∆)) = e(∆), then

4.2 Right classification of univariate power series

99

1. q(∆) = m(∆). 2. k(∆) = 0 and d(∆) = m(∆). (c) If q(∆) = q(∆0 ) =: q and ∆ ∩ N e0 > 1. Then k = k∆ (n) for some n ∈ ∆ and e(n) ≥ 2. It yields that k = k∆ (n) = d

q−n q−n q e≤d 2 e k∆ (q0 ). Using the same argument as in Case 2 we get q ]Λ(∆) ≥ b c. p 0 Case 4: e(m) > e0 = 1 and k = k∆ (q0 ) and k(∆0 ) ≥ b q−q c. p Then m q q ]Λ0 (∆) ≤ b 2 c − b 2 c ≤ b 2 c − 1 p p p and d q d q ]Λ01 (∆) = k − b c − b c + b 2 c − b 2 c p p p p q q0 d q = b c − + b 2c − b 2c + 1 p p p p

d q0 since k − b c = − + 1 according to Remark 4.2.1(e). Hence p p q q0 d ]Λ(∆) = ]Λ0 (∆) + ]Λ01 (∆) ≤ b c − + b 2 c. p p p 0 Moreover since k(∆0 ) ≥ b q−q c, p

d

q0 − n q − q0 e≥b c e(n) p −p p

for some n ∈ ∆0 with e(n) > 1. It follows easily that pq0 ≥ d and hence q q0 d q ]Λ(∆) ≤ b c − + b 2 c ≤ b c. p p p p 0 Case 5: e(m) > e0 = 1 and k = k∆ (q0 ) and k(∆0 ) < b q−q c. p Then 2q0 − m m 2q0 ]Λ0 (∆) ≤ b c − b 2 c ≤ b 2 c − 2. 2 p p p and d q q q0 ]Λ001 (∆) = k − b c − b c = b c − + 1. p p p p Hence q q0 2q0 q ]Λ(∆) = ]Λ0 (∆) + ]Λ001 (∆) < b c − + b 2 c ≤ b c. p p p p


101

Note that if mt(f ) = 0 then mt(f − f (0)) > 0. Applying the results from mt(f ) > 0 to f − f (0) we obtain that f ∼r f (0) + g, where g is a normal form of f − f (0) (cf. Theorem 4.2.7). From now we always assume that mt(f ) > 0. Let supp(f ) be the support of f . Then mt(f ) = m(supp(f )). We denote ¯ ) := (a) e(f ) := e(supp(f )), q(f ) := q(supp(f )), k(f ) := k(supp(f )), d(f ¯ ¯ ) := Λ(supp(f ¯ d(supp(f )), d(f ) := d(supp(f )) and Λ(f )). (b) Λ(f ) := {n · pe(f ) | n ∈ Λ(supp(f¯))} with f¯ defined as in Remark 4.2.3(b) below. Note that if e(f ) = 0 then Λ(f ) = Λ(supp(f )). Remark 4.2.3. (a) The above numbers m, e, q, k, d¯ and d are invariant w.r.t. right equivalence. P (b) Let f = n≥1 cn xn ∈ K[[x]] and let f¯(x) =

e(f )

X

cn xn/p

.

n≥mt(f ) e(f ) Then f¯ ∈ K[[x]], f (x) = f¯(xp ) and e(f¯) = 0. Moreover, k(f ) = k(f¯) and if ¯ ), d(f ) then ζ(f ) denotes one of mt(f ), e(f ), q(f ), d(f

ζ(f ) = pe(f ) ζ(f¯). (c) Note that µ(f ) < ∞ if and only if e(f ) = 0 and then q(f ) = µ(f ) + 1 and ¯ ) = 2µ(f ) − mt(f ) + 2. Moreover, f is then right d(f ¯ )-determined by d(f [BGM12, Thm. 2.1]. In Proposition 4.2.6 below we give a better bound for determinacy of a univariate power series. Lemma 4.2.4. If e(mt(f )) = e(f ) then f ∼r xmt(f ) . In particular if p - mt(f ) then f ∼r xmt(f ) . e(f ) Proof. By Remark 4.2.3, there exists f¯ ∈ K[[x]] such that f (x) = f¯(xp ) and e(f¯) = 0. This implies that µ(f¯) = q(f¯) and then µ(f¯) = mt(f¯)−1 since e(mt(f )) = e(f ). It follows from [BGM12, Thm. 2.1] that f¯ is right (mt(f¯) + 1)-determined and hence ¯ ¯ f¯ ∼r cxmt(f ) ∼r xmt(f ) .

In fact, in this case an inductive proof as in the case of characteristic 0 works. Lemma 4.2.5. A univariate power series f ∈ K[[x]] is right equivalent to xmt(f ) +

X ¯ ) n∈Λ(f

for suitable λn ∈ K.

λn xn ,

102


Proof. It is sufficient to prove the lemma for the case that e(f ) = 0 by Remark 4.2.3. We decompose f = f0 + f1 with X X f0 := ci xi and f1 := cn x n . e(i)=0

e(n)>0

Then mt(f0 ) = q(f ) and e(mt(f0 )) = e(f0 ) = 0 and hence f0 ∼r xq(f ) by Lemma 4.2.4. That is, ϕ(f0 ) = xq(f ) for some coordinate change ϕ ∈ Aut(K[[x]]). It yields that g := ϕ(f ) = ϕ(f0 ) + ϕ(f1 ) = xq(f ) + ϕ(f1 ). ¯

This implies that f ∼r g ∼r j d(g) (g) due to Remark 4.2.3(c). It is easy to see that ¯ j d(g) (g) has a form as required. The next proposition is the second key step in the classification. Proposition 4.2.6. Let f ∈ K[[x]] and d = d(f ). Let g ∈ K[[x]] be such that e(f ) = e(g) and j d (f ) = j d (g). Then f ∼r g. In particular, if µ(f ) < ∞ then f is right d-determined. Proof. The proof will be divided into two steps. Step 1: e := e(f ) = 0. Since j d (f ) = j d (g), there exists l with q + l ≥ d + 1 and bq+l 6= 0 s.t. g − f = bq+l xq+l mod xq+l+1 , with d := d(f ) = d(g), q := q(f ) = q(g). We will construct inductively a finite ¯ ) − q − l + 1 with f0 = f such that fi ∼r f for all sequence of fi , i = 0, . . . , N := d(f i and that g − fi = 0 mod xq+l+i . If we succeed then by Remark 4.2.3(c) we have f ∼r fN ∼r g. In fact since q + l ≥ d + 1, q−n e| mt(f ) ≤ n < q, n ∈ supp(f )}. l ≥ k(f ) = max{ d e(n) p −1 Considering the coordinate change ϕ1 (x) = x + ul+1 xl+1 with ul+1 a solution of the following equation: X e(n) e(n) (n/pe(n) )p cn X p + bq+l = 0 qcq X + q−n =l pe(n) −1

and setting f1 := ϕ1 (f ) we get g − f1 = 0 mod mq+l+1 and hence we obtain by induction a finite sequence of (fi )i=0,...,N as required. Step 2: e := e(f ) > 0. Taking f¯ ∈ K[[x]] and g¯ ∈ K[[x]] such that f (x) = e e f¯(xp ), g(x) = g¯(xp ) as in Remark 4.2.3 we have e(f¯) = e(¯ g ) = 0, q(f¯) = q(¯ g ) = q/pe , d¯ := d(f¯) = d(¯ g ) = d/pe . ¯ ¯ Since j d (f ) = j d (g), j d (f¯) = j d (¯ g ) and hence f¯ ∼r g¯ by the first step. This implies that f ∼r g with the same coordinate change.


103

Theorem 4.2.7 (Normal form of univariate power series). With f , mt(f ) and Λ(f ) as above, we have X f ∼r xmt(f ) + λ n xn n∈Λ(f )

for suitable λn ∈ K. Proof. It is sufficient to prove the theorem for the case that e(f ) = 0 by Remark 4.2.3. We proceed step by step as in definition of Λ(∆) with ∆ := supp(f ). Case 0: e(m) = 0 with m := mt(f ). The claim follows from Lemma 4.2.4. Case 1: e(m) > 0 and if e(m) = e(f0 ) =: e0 with ∆0 := {n ∈ ∆ | n < q} and f0 :=

X

cn x n .

n∈∆0

Applying Lemma 4.2.4 to f0 we obtain that f0 ∼ xm , i.e. there exists a coordinate change ϕ such that ϕ(f0 ) = xm and then ϕ(f ) = xm + ϕ(f1 ) with f1 := f − f0 =

X

cn x n .

n≥q

We write ϕ(f1 ) :=

X

bn xn , bq 6= 0.

n≥q

Since k := k(f ) = k∆ (m) we have m + lpe(m) < q + l for all l < k. This allows us to eliminate inductively all coefficients bn in ϕ(f1 ) with e(n) ≥ e(m) and q ≤ n ≤ d by a suitable coordinate change (e.g. if n1 is the minimum for which e(n1 ) ≥ e(m), q ≤ n1 ≤ d and bn1 6= 0, then the coordinate change ϕ1 (x) = x + ul+1 xl+1 with l =

n1 − m and ul+1 a solution of the equation: pe(m) m/pe(m)

pe(m)

e(m)

Xp

+ bn1 = 0,

makes the coefficient of xn1 vanishing, and the terms of exponents less than n1 do not change). This completes the proof. Case 2: e(m) > e0 > 1. Then applying Lemma 4.2.5 to f0 we get X f 0 ∼ r xm + λ n xn ¯ 0) n∈Λ(f

104


for suitable λn , i.e. X

ϕ(f0 ) = xm +

λn xn

¯ 0) n∈Λ(f

for some coordinate change ϕ. Then X

ϕ(f0 ) = xm +

λ n xn

mod xq

n∈Λ0 (∆)

and hence ϕ(f ) ∼r xm +

X

X

λ n xn +

n∈Λ0 (∆)

λn xn

n∈Λ1 (∆)

by Proposition 4.2.6. This proves the claim. Case 3: e(m) > e0 = 1 and if k(f ) > k∆ (q0 ) with q0 := q(∆0 ) = min{n ∈ ∆0 | e(n) = e0 }. Using the same argument as in Case 2 gives us the claim. Case 4,5: If e(m) > e0 = 1 and k(∆) = k∆ (q0 ). This is done by the same method as in Case 1 and Case 2.

4.2.2

Right modality

Theorem 4.2.8. Let charK = p > 0. Let f ∈ hxi ⊂ K[[x]] be a univariate power series such that its Milnor number µ := µ(f ) is finite. Then R-mod(f ) = bµ/pc. Proof. We first prove the inequality R-mod(f ) ≤ bµ/pc. Indeed, let k = 2µ(f ) and let I := {∆ ⊂ N| q(f ) ∈ ∆, 1 ≤ n ≤ k ∀n ∈ ∆}. With the unfolding ft (x) be as above, by the upper semicontinuity of the Milnor number (Lemma 3.4.3), there exists an open subset U of t0 ∈ Aµ such that µ(ft ) ≥ µ(f ) for all t ∈ U . This implies that supp(ft ) ∈ I for all t ∈ U . We can easily verify that the finite set of families X (n) hs∆ (x) := xm(∆) + s∆ x n , ∆ ∈ I n∈Λ(∆)

over A∆ ≡ Al∆ with l∆ = ]Λ(∆), satisfies the assumption of Proposition 3.4.17 and hence R-mod(f ) ≤ max ]Λ(∆) ≤ bq/pc = bµ/pc ∆∈I

by proposition 4.2.2. In oder to prove the other inequality we consider the two following cases. Case 1: mt(f ) = p.


105

Then q := q(f ) = µ(f ) + 1, Λ(f ) = {n ≥ q|e(n) = 0} and ]Λ(f ) = bq/pc due to Proposition 4.2.2. It follows from Proposition 4.2.7 that X f ∼r fλ := xp + λ n xn n∈Λ(f )

for suitable λn ∈ K. Let us show that if fλ ∼r fλ0 then (λn )n∈Λ(f ) = (λ0n )n∈Λ(f ) . In fact, since fλ ∼r fλ0 , there exists a coordinate change ϕ := ax + al xl+1 + . . . such that ϕ(fλ ) = fλ0 . Then ap = 1 and therefore a = 1. If l < k(f ) then the point p(l + 1) ∈ supp(ϕ(fλ )) but p(l + 1) 6∈ supp(fλ0 ), that is ϕ(fλ ) 6= fλ0 . The contradiction gives us l ≥ k(f ). It then follows from elementary calculations that j d (fλ ) = j d (ϕ(fλ )) = j d (fλ0 ) and hence (λn )n∈Λ(f ) = (λ0n )n∈Λ(f ) . This implies that R-mod(f ) ≥ ]Λ(f ) = bq/pc = bµ/pc. Case 2: mt(f ) > p. By the upper semicontinuity of the right modality (Proposition 3.4.7) one has R-mod(f ) ≥ R-mod(ft ) with ft = f + t · xp for all t in some neighbourhood W of 0 in A1 . Take a t0 ∈ W \ {0} then R-mod(ft0 ) = bµ/pc by the first step and hence R-mod(f ) ≥ R-mod(ft0 ) = bµ/pc.

Remark 4.2.9. We have R-mod(f ) ≥ ]Λ(f ) by Theorem 4.2.8 and Proposition 4.2.2 with equality if mt(f ) ≤ p. Moreover, if mt(f ) = p, then fλ ∼r fλ0 for λ, λ0 ∈ Λ(f ) implies λ = λ0 , which follows from the proof of Theorem 4.2.8. The example f = xp+1 with R-mod(f ) = 1 but Λ(f ) = ∅ shows that a strict inequality R-mod(f ) > ]Λ(f ) can happen. Note that if µ := µ(f ) is finite then the system {x, . . . , xµ } is a basis of m/m·j(f ). Proposition 3.4.16 shows that X ft (x) = f + t i · xi is the semiuniversal unfolding (with trivial section) of f over Aµ . We define Σµ := {t ∈ Aµ | µ(ft ) = µ} the µ-constant stratum of the unfolding ft and call its dimension dim Σµ the inner modality of f . Corollary 4.2.10. The modality and the inner modality coincide. That is, for any f ∈ hxi ⊂ K[[x]] with the Milnor number µ < ∞. Then R-mod(f ) = dim Σµ .

106


Proof. For each t = (t1 , . . . , tµ ) ∈ Aµ , if the set Nt := {i = 1, . . . , µ | ti 6= 0, e(i) = 0} is not empty, then µ(ft ) = n − 1 < µ with n := min{i | i ∈ Nt }. This implies that Σµ = {t = (t1 , . . . , tµ ) ∈ Aµ | ti = 0, ∀e(i) = 0}. µ It yields dim Σµ = b c and hence R-mod(f ) = dim Σµ by Theorem 4.2.8. p

4.3

Right simple singularities

In this section we classify the right simple singularities f ∈ K[[x1 , . . . , xn ]] for K an algebraically closed field of characteristic p > 0. In contrast to char(K) = 0, where the classification of right simple and contact simple singularities coincides, the classification is very different in positive characteristic. For example, for every p > 0, there are only finitely many classes of right simple singularities and for p = 2 only the A1 -singularity in an even number of variables is right simple. The classification of right simple singularities is summarized in Proposition 4.3.1 and Theorems 4.3.2 and 4.3.3. Proposition 4.3.1. Let charK = p > 0. A univariate singularity f ∈ K[[x]] with finite Milnor number µ, is right simple if and only if µ < p, and then f ∼r xµ+1 . Proof. Follows from Lemma 4.2.4 and Theorem 4.2.8. Theorem 4.3.2. Let p = char(K) > 2. (i) A plane curve singularity f ∈ m2 ⊂ K[[x, y]] is right simple if and only if it is right equivalent to one of the following forms Name Ak Dk E6 E7 E8

Normal form x2 + y k+1 1≤k ≤p−2 2 k−1 x y+y 4≤k
3, let mt(f ) = 3 and and f3 be the tangent cone (i.e. the homogeneous component of degree 3) of f . Let r(f3 ) be the number of linear factors of f3 . (i) If r(f3 ) ≥ 2 then f ∼r x2 y + g(y) with mt(g) = µ − 1 ≥ 3. If additionally 4 ≤ µ < p, then f ∼r Dµ and f is right simple. (ii) If r(f3 ) = 1, p = 5 and 6 ≤ µ ≤ 7 then f ∼r Eµ and f is right simple. (iii) If r(f3 ) = 1, p > 5 and 6 ≤ µ ≤ 8 then f ∼r Eµ and f is right simple. Proof. Step 1: We prove the first claim for each statement (i), (ii) and (iii). (i) If r(f3 ) = 3 then we can easily show that f3 ∼r D4 := y(x2 + y 2 ) and that m4 ⊂ m2 · j(D4 ). It follows from Theorem 1.1.10 that D4 is right 4-determined. This implies that f ∼r D4 . If r(f3 ) = 2 then f3 ∼r x2 y. So we may assume that f3 = x2 y. By Corollary 1.2.24 f can be factorized as f = f¯1 · f¯1 with (f¯1 )1 = y and (f¯2 )2 = x2 , where (f¯1 )1 resp. (f¯2 )2 is the tangent cone of f¯1 resp. f¯2 . We may assume that f¯1 = y and then f = y · f¯2 with f¯2 = x2 + x · h1 (x, y) + h2 (y). The coordinate change x 7→ x − 21 h1 (x, y); y 7→ y suffices to increase the multiplicity of h1 (x, y). Continuing by induction we get a coordinate change x 7→ Φ(x); y 7→ y such that f¯2 (Φ(x), y) = x2 + h(y). Hence f ∼r x2 y + g(y). (ii), (iii): cf. [GLS06, Thm. I.2.53]. Note that the proof in [GLS06] was given for the complex case but works in positive characteristic.

108


Step 2: Let f be Dµ , E6 , E7 or E8 (if p > 5) and let ft be the semiuniversal unfolding (with trivial section) of f at t0 ∈ T (see, Prop. 3.4.16). By the upper semi-continuity of the Milnor number (cf. Lem. 3.4.3), there exits an open subset W ⊂ T of t0 such that µ(ft ) ≤ µ(f ) for all t ∈ W . It follows from Proposition 4.3.4 and the first step that for each t ∈ W , ft is right equivalent to one of the following singularities x, Ai , Di , E6 , E7 , E8 with i ≤ µ. Hence f is right simple due to Corollary 3.4.18. Proposition 4.3.6. Let mt(f ) = 3. Let r(f3 ) be the number of linear factors of f3 . Then f is not right simple if (i) either p = 3; (ii) or p > 3, r(f3 ) ≥ 2 and µ ≥ p; (iii) or p > 5, r(f3 ) = 1 and µ > 8; (iv) or p = 5, r(f3 ) = 1 and µ ≥ 8. Proof. (i) We consider the unfolding F (x, y, t) = f + t · x2 of f at 0 over K. Since mt(f ) = 3 and since p = 3, it is easy to see that µ(ft ) > 2 for all t 6= 0. Proposition 4.3.4(ii) yields that ft , t 6= 0 is not right simple and hence neither is f by Proposition 3.4.7. (ii) By Proposition 4.3.5(i), f ∼r x2 y + g(y) with mt(g) = µ − 1. It suffices to show that x2 y + g(y) is not right simple. We write g(y) = a · y µ−1 + . . . with a 6= 0 and consider the unfolding ( x2 y + g(y) + tx2 + ty p ft := F (x, y, t) = x2 y + g(y) + at2 x2 + atxy (p−1)/2

if µ > p if µ = p

of x2 y + g(y) at 0 over K. It is easy to see that µ(ft ) ≥ p for all t 6= 0. It follows from Proposition 4.3.4 that ft , t 6= 0 is not right simple and hence neither is x2 y + g due to Proposition 3.4.7. (iii) This is done by the same argument as in [GLS06, Thm. I.2.55(2)(ii)]. (iv) Since r(f3 ) = 1 and µ ≥ 8, using the same argument as in [GLS06, Thm. I.2.53] we get f ∼r g := x2 y + αy 5 + βxy 4 + h(x, y) with α, β ∈ K and h ∈ m6 . Consider the unfolding gt := G(x, y, t) = g(x, y) + t · xy 4 of g at 0 over K and assume that gt ∼r gt0 , i.e. there exists an automorphism Φ : K[[x, y]] −→ K[[x, y]] X x 7→ ϕ = aij xi y j X y 7→ ψ = bij xi y j such that gt (x, y) = gt0 (ϕ, ψ). By a simple calculation we conclude that (β + t)3 = (β + t0 )3 and hence, for fixed t, gt ∼r gt0 for at most three values of t0 . It follows from Corollary 3.4.18 that g is not right simple and hence neither is f .

4.3 Right simple singularities

109

Proof of Theorem 4.3.2(i). It follows from Propositions 3.4.19, 4.3.4, 4.3.5 and 4.3.6.

4.3.2

Right simple hypersurface singularities in characteristic > 2

The aim of this paragraph is to prove Theorem 4.3.2(ii). K[[x1 , . . . , xn ]]. We denote by H(f ) :=

Let f ∈ K[[x]] =

∂ 2f (0) i,j=1,...,n ∈ Mat(n × n, K) ∂xi ∂xj

the Hessian (matrix) of f and by crk(f ) := n − rank(H(f )) the corank of f . Lemma 4.3.7 (Right splitting lemma in characteristic different from 2). If f ∈ m2 ⊂ K[[x]], char(K) > 2, has corank crk(f ) = k ≥ 0, then f ∼r g(x1 , . . . , xk ) + x2k+1 + . . . + x2n with g ∈ m3 . g is called the residual part of f , it is uniquely determined up to right equivalence. Proof. cf. [GLS06, Thm. I.2.47]. The proof in [GLS06] was given for K = C but works in characteristic different from 2. Lemma 4.3.8. Let p = char(K) > 2 and let fi (x1 , . . . , xn ) = x2n + fi0 (x1 , . . . , xn−1 ) ∈ m2 ⊂ K[[x1 , . . . , xn ]], i = 1, 2. Then f1 ∼r f2 if and only if f10 ∼r f20 . Proof. The direction, f10 ∼r f20 ⇒ f1 ∼r f2 is obvious. We now show the other one, f1 ∼r f2 ⇒ f10 ∼r f20 . First, assume that f1 ∼r f2 . Then crk(f1 ) = crk(f2 ) := k and therefore crk(f10 ) = crk(f20 ) = k. It follows from Lemma 4.3.7 that fi0 ∼r gi (x1 , . . . , xk ) + x2k+1 + . . . + x2n−1 , i = 1, 2. and hence fi ∼r gi (x1 , . . . , xk ) + x2k+1 + . . . + x2n−1 + x2n . This implies g1 (x1 , . . . , xk ) + x2k+1 + . . . + x2n ∼r g2 (x1 , . . . , xk ) + x2k+1 + . . . + x2n since f1 ∼r f2 . The uniqueness of gi shows that g1 ∼r g2 , i.e. there exists an automorphism Φ0 ∈ AutK (K[[x1 , . . . , xk ]]) such that Φ0 (g1 ) = g2 . The automorphism Φ : K[[x1 , . . . , xn−1 ]] −→ K[[x1 , . . . , xn−1 ]] xi 7→ Φ0 (xi ), i = 1, . . . , k xj 7→ xj , j = k + 1, . . . , n − 1

110


yields that g1 (x1 , . . . , xk ) + x2k+1 + . . . + x2n−1 ∼r g2 (x1 , . . . , xk ) + x2k+1 + . . . + x2n−1 and hence f10 ∼r f20 . This completes the proof. Lemma 4.3.9. Let p = char(K) > 2, n ≥ 2, and let f (x1 , . . . , xn ) = x2n + f 0 (x1 , . . . , xn−1 ) ∈ m2 ⊂ K[[x1 , . . . , xn ]] be such that µ(f ) < ∞. (i) Let F 0 (x0 , t) ∈ hx1 , . . . , xn−1 i ⊂ O(T )[[x1 , . . . , xn−1 ]] be an unfolding of f 0 at t0 over an affine variety T and let F (x, t) = x2n + F 0 (x0 , t). Then R-modF (f ) = R-modF 0 (f 0 ). (ii) We have R-mod(f ) in K[[x1 , . . . , xn ]] = R-mod(f 0 ) in K[[x1 , . . . , xn−1 ]]. Proof. Let k be sufficiently large for f and for f 0 w.r.t. R. Let m0 be the maximal ideal in K[[x0 ]] and let Jk := K[[x0 ]]/m0k+1 . (i) Consider three morphisms hk : T −→ Jk t 7→ j k ft

and

h0k : T −→ Jk0 t 7→ j k ft0 .

and p : Jk → Jk0 the natural projection. Then h0k = p ◦ hk and 0−1 0 h−1 k (R · hk (t)) = hk (R · hk (t))

by Lemma 4.3.8. It follows from Corollary 3.3.6 that R-modhk (t0 ) = R-modh0k (t0 ), and hence R-modF (f ) = R-modF 0 (f 0 ). (ii) Let {g10 (x0 ), . . . , gl0 (x0 )} be a system of generators of m0 /m0 · j(f 0 ). Then {xn , g1 (x), . . . , gl (x)} with gi (x) = gi0 (x0 ), is a system of generators of m/m · j(f ). Proposition 3.4.16 yields that 0

0

0

F (x , t) = f +

l X

ti gi0 (x0 )

resp. F1 (x, t) = f +

i=1

l X

ti gi (x) + t0 xn

i=1

is a right complete unfoldings of f 0 resp. f . Consider the unfolding F =f+

l X

ti gi (x)

i=1

of f . It is not difficult to see that R-modF1 (f ) = R-modF (f ). This implies, by (i), that R-modF1 (f ) = R-modF (f ) = R-modF 0 (f 0 ) and hence R-mod(f ) = R-mod(f 0 ) due to Proposition 3.4.14(ii).

4.3 Right simple singularities

111

Proof of Theorem 4.3.2(ii). The “if”-statement follows from Theorem 4.3.2(i) and Lemma 4.3.9. We now consider any simple singularity f ∈ m2 ⊂ K[[x]]. Then by the splitting lemma, f ∼r f 0 (x1 , . . . , xk ) + x2k+1 + . . . + x2n with f 0 ∈ hx1 , . . . , xk i3 and k = crk(f ). Again by Lemma 4.3.9, R-mod(f 0 ) = R-mod(f ) = 0. It follows from Proposition 3.4.19 that m+k−1 0 = R-mod(f ) ≥ − k2, m 0

where m = mt(f 0 ) ≥ 3. This implies that k ≤ 2, i.e. f ∼r g(x1 , x2 ) + x23 + . . . + x2n , for some simple singularity g ∈ K[[x1 , x2 ]]. The proof thus follows from Proposition 4.3.1, Theorem 4.3.2(i) and Lemma 4.3.8.

4.3.3

Right simple hypersurface singularities in characteristic 2

Let p = char(K) = 2 and let n ≥ 2. Lemma 4.3.10 (Right splitting lemma in characteristic 2). Let f ∈ m2 ⊂ K[[x]] = K[[x1 , . . . , xn ]], n ≥ 2. Then there exists an l, 0 ≤ 2l ≤ n such that f ∼r x1 x2 + x3 x4 + . . . + x2l−1 x2l + g(x2l+1 , . . . , xn ) with g ∈ hx2l+1 , . . . , xn i3 or g ∈ x22l+1 + hx2l+1 , . . . , xn i3 if 2l < n. g is called the residual part of f , it is uniquely determined up to right equivalence. Proof. It follows easily from [GK90, Lemma 1 and 2]. Lemma 4.3.11. Let µ(f ) < ∞ and f (x1 , . . . , xn ) = xn−1 xn + f 0 (x1 , . . . , xn−2 ) ∈ m2 ⊂ K[[x1 , . . . , xn ]]. Then R-mod(f ) in K[[x1 , . . . , xn ]] = R-mod(f 0 ) in K[[x1 , . . . , xn−2 ]]. Proof. By using the same argument as in the proof of Lemma 4.3.9. Remark 4.3.12. Since µ(x1 x2 ) = 1, x1 x2 ∈ K[[x1 , x2 ]] is right 2-determined and any unfolding of x1 x2 is either right equivalent to itself or smooth. Hence x1 x2 is right simple. Proposition 4.3.13. Let f ∈ m2 ⊂ K[[x]] with µ(f ) < ∞. Then f is not right simple if

112


(i) either f = x21 + g(x1 , . . . , xn ) ∈ K[[x]] with g ∈ m3 , (ii) or f ∈ m3 . Proof. (i) Let k ≥ 3 be sufficiently large for f w.r.t. R and let X := m2 /mk+1 . Then R-mod(f ) = Rk -mod(j k f ). Let Y := x21 + m3 /mk+1 ⊂ Jk , Y 0 := x21 + m3 /m4 and let H := {Φ ∈ Rk |Φ(x1 ) = x1 }, H0 := {Φ ∈ R1 |Φ(x1 ) = x1 }. Then H (resp. H0 ) acts on Y (resp. Y 0 ) by (Φ, y) 7→ Φ(y) and we have i−1 (Rk · i(y)) = H · y ⊂ p−1 (H0 · p(y)) ∀y ∈ Y 0 with the inclusion i : Y ,→ X and the projection p : Y Y 0 . It follows from Proposition 3.2.7 that Rk -mod(y) ≥ H-mod(y) ≥ H0 -mod(p(y)), ∀y ∈ Y. Moreover, Proposition 3.2.4 yields that n+2 H -mod(p(y)) ≥ dim Y − dim H = − n(n − 1) ≥ 1. 3 0

0

0

This implies that Rk -mod(y) ≥ 1 for all y ∈ Y and hence R-mod(f ) ≥ 1. (ii) By (i), ft is not right simple for all t 6= 0, where ft (x) := F (x, t) = f (x) + tx21 is an unfolding of f at 0 over K. Hence Proposition 3.4.7 yields that f is not right simple. Proof of Theorem 4.3.3. The “if”-statement is obvious. Now, take a right simple singularity f ∈ m2 ⊂ K[[x]]. Then mt(f ) = 2 by Proposition 3.4.19. The splitting lemma (Lemma 4.3.10) yields that f is right equivalent to x1 x2 + x3 x4 + . . . + x2l−1 x2l + g(x2l+1 , . . . , xn ) with g ∈ hx2l+1 , . . . , xn i3 or g ∈ x22l+1 +hx2l+1 , . . . , xn i3 if 2l < n. Combining Lemma 4.3.11 and Proposition 4.3.13 we obtain that 2l = n, which proves the theorem.

Chapter 5 Drozd-Greuel’s Ideal-Unimodal Plane Curve Singularities In this chapter, R denotes a complete local noetherian ring without nilpotent element and m its maximal ideal such that the residue field K = R/m is algebraically closed.

5.1

Definition

Let R be a local ring and m its maximal ideal. We denote Q the full ring of fractions of R. R0 the normalization of R. Ri := mi R0 + R (a local ring for i > 0). mi := mi R0 + m the maximal ideal of Ri for i > 0. d(M ) := dimK (M/mM ), the minimal number of generators of an R-module M. di := d(Ri ). Definition 5.1.1. R is said to be a curve singularity, if it is a K-algebra of Krull dimension 1. L Let R be a curve singularities. We may assume that R0 = ri=1 K[[t]], where r is the number of branches of R. Then L • R1 = K + ri=1 tmi K[[t]], where mi is the multiplicity of i-th branch; d1 = d(R1 ) = dimK (R0 /R1 ) + 1. • In general, we have R0 ⊃ R1 ⊃ · · · ⊃ R; Ri+1 = K + mRi . Hence di = d(Ri ) = dimK (Ri−1 /Ri ) + 1. L • Let x = (x1 , . . . , xr ) ∈ R ⊂ R0 = ri=1 K[[t]]. Then the (multi) valuation of x, v(x), is defined by v(x) := (ord(x1 ), . . . , ord(xr )) ∈ (Z ∪ {∞})r . 113

114

Chapter 5. Drozd-Greuel’s Ideal-Unimodal Plane Curve Singularities

• If R = K[[x, y]]/hf i is a plane curve singularities and let {x, y} a system of generator Lr of m. Let x(t) resp. y(t) be the image of x resp. y by the inclusion R ,→ i=1 K[[t]]. Then ψ(t) = (x(t), y(t)) is a parametrization of some g being contact equivalent to f . According to Drozd and Greuel [DG98] we make the following definition. Definition 5.1.2. A plane curve singularity with complete local ring R ⊂ R0 is called ideal-unimodal (IUS) if its maximal ideal admits generators x, y whose valuation satisfies the condition in Table 5.1. Moreover, we say that R is of type E, T, W or Z, if R belongs to the corresponding class.

Type

r

v(x)

v(y)

E

1 2

(3) (1, 2)

(l) (∞, l)

T

W

Z

3 (1, 1, 1) 2 (2, k) 3 (1, 1, k) 4 (1, ∞, 1, k) 1 (4)

(∞, l, k) (l, 2) (∞, l, 2) (∞, 1, l, 1) (l)

2 2

(1, 3) (2, 2)

(∞, l) (3, l)

3 2 3

(1, 1, 2) (1, l) (1, ∞, 2)

(∞, l, 3) (∞, 3) (∞, 1, l)

4

(1, ∞, 1, 1)

(∞, 1, l, k)

Condition

Name

l = 7, 8, 10, 11 l = 4, 5, 6, 7

E12 , E14 , E18 , E20 E2,2q−1 , E13 , E3,2q−1 (q ≥ 1), E19 l = 2, 3, k ≥ l El,2(k−1) l, k odd, lk > 4 Tk+2,l+2,2 k odd, lk ≥ 2 Tk+2,2l+2,2 lk ≥ 1 T2k+2,2l+2,2 ] l = 5, 6, 7 (∗) W12 , W1,2q−1 (q ≥ 1), W18 l = 4, 5 W13 , W17 ] l = 3 (∗) W1,0 , W1,2q , (q ≥ 1) l ≥ 5, odd W1,l−3 l≥2 W1,2l−3 l = 4, 5, 7, 8 Z11 , Z13 , Z17 , Z19 l = 2, 3, 4, 5 Z0,2q−1 , Z12 , Z1,2q−1 (q ≥ 1), Z18 l = 1, 2, k ≥ l Zl−1,2(k−1) Table 5.1:

r: number of branches. v(x), v(y): valuation of x, y respectively. (∗) If char(K) = 2, extra conditions in case W are required: if r = 1 then l = 5, if r = 2, v(x) = (2, 2) and l = 3 then (x2 − (y3 6∈ t7 R0 . ] (r = 1 or r = 2).) (This excludes W18 (r = 1) and all W1,q Remark 5.1.3. 1. The index q is not specified in [DG98], it depends on the deltainvariant of the corresponding singularity. More precisely, δ(E2,2q−1 ) = q + 5 and δ(E3,2q−1 ) = q + 8. ] ] δ(W1,0 ) = 8, δ(W1,2q−1 ) = q + 7 and δ(W1,2q ) = q + 8. δ(Z0,2q−1 ) = q + 5, δ(Z1,2q−1 ) = q + 8. 2. The name in the last column of Table 5.1 corresponds to a whole class of singularities (not just to an individual singularity).

5.1 Definition

115

3. A change of coordinates and a reparametrization of a plane curve singularity does not change the conditions characterizing the classes in Table 5.1. Hence the classes E, T, W or Z (with fixed indices) in Table 5.1 are invariant under parametrization equivalence. 4. As we shall see in Proposition 5.2.1, the classes in Table 5.1 are not disjoint. From now on, we shall consider only full R-ideals, i.e. ideals I, such that QI = Q (later we omit the epithet“ful”). Definition 5.1.4. Let (X, OX ) be an algebraic affine variety over K and I an R ⊗ OX -ideal sheaf, such that QI = Q ⊗ OX (the tensor product is over K). Call I a family of ideals with base X if it flat over OX and if I/sI is OX -flat for each non-zero divisor s ∈ R. Remark 5.1.5. 1. Let V be a K-vector space. We call the set Gr(d, V ) := {W |W is a K − subspace of V of codimension d} a Grassmanian variety. 2. We consider the subvariety B(d) := {W ∈ Gr(d, R0 /R)|W is a R submodule of R0 /R} of Gr(d, R0 /R) and denote by I(d) the pre-image in R0 ⊗ OB(d) of the canonical locally free sheaf of corank d on B(d). Then I(d) is a family of ideals with base B(d). Definition 5.1.6. Let B(d, i) the subset of B(d) consisting of points x such that the set {y ∈ B(d)|I(d)(x) ' I(d)(y)} (which is locally closed) has dimension i and call par(1, R) = max{dim B(d, i) − i} d,i

the number of parameters for R-ideals. Definition 5.1.7. We say that a ring R0 dominates R if R ⊂ R0 ⊂ R0 . In this case, evidently, par(1, R0 ) ≤ par(1, R). Theorem 5.1.8 (Drozd and Greuel). Let R be a curve singularity. The following conditions are equivalent: 1. par(1, R) ≤ 1. 2. R dominates a simple or ideal-unimodal plane curve singularity. 3. (a) d(R0 ) ≤ 4; (b) d(R1 ) ≤ 3; (c) d(R2 +eR) ≤ 3 for any idempotent e ∈ R, such that d(eR0 ) = 1 (provided it exists); (d) if d(R0 ) = 3, then d(R3 ) ≤ 2;

116

Chapter 5. Drozd-Greuel’s Ideal-Unimodal Plane Curve Singularities ˜ I˜ 6' A0 , where I˜ := t2 mR0 + m; R ˜ := EndI˜ := (e) if char(K) = 2, then R/ ˜ ˜ {ϕ ∈ HomR0 (I, I)| ϕ(x) = sx, s ∈ R0 } the endomorphism ring; A0 the 4-dimensional K-algebra having a basis {1, a, b, ab} with a2 = b2 = 0.

Proof. cf. [DG98, Theorem 2.1]. Corollary 5.1.9 (Drozd and Greuel). i) Let R be a plane curve singularity. Then R is simple or ideal-unimodal if and only if the equivalent conditions of theorem 5.1.8 are satisfied. ii) If char(K) = 0, then R dominates a simple or strictly unimodal plane curve singularity if and only if the equivalent conditions of theorem 5.1.8 are satisfied. Proof. cf. [DG98, Corollary 2.2]. Proposition 5.1.10 (Drozd and Greuel). Let char(K) = 0 and R be a plane curve singularity. Then R is ideal-unimodal if and only if it is (contact) strictly unimodal. Proof. cf. [DG98, Proposition 2.3]. Corollary 5.1.11. Let char(K) = 0 and R be a plane curve singularity. Then R is contact equivalent to a singularity of Table 5.1 if and only if it is (contact) strictly unimodal. Proof. It is straightforward from propositions 5.1.10 and 1.2.19.

5.2

A disjoint list of IUS

In this section we give a table of IUS which all classes are mutually disjoint. We first show that the classes in Table 5.1 are not disjoint. In the following for each integer n, O(n) denotes a power series in K[[t]] of order at least n. Proposition 5.2.1. The following classes of singularities coincide. (i) E2,2q−1 = T2q+3,3,2 ; q ≥ 1; (r = 2) (ii) E2,2(l−1) = T3,2l+2,2 ; l ≥ 2; (r = 3). (iii) Tk+2,l+2,2 = Tl+2,k+2,2 and T2k+2,2l+2,2 = T2l+2,2k+2,2 . (iv) Z0,2q−1 = T2q+3,4,2 ; q ≥ 1; (r = 3). (v) Z0,2(k−1) = T2k+2,4,2 ; k ≥ 1; (r = 4). Proof. (i) We first show that E2,2q−1 ⊂ T2q+3,3,2 . Let R be a singularity of type E2,2q−1 and x, y a generators of its maximal ideal satisfying the condition as in Table 5.1. Let f ∈ K[[x, y]] be such that R = K[[x, y]]/hf i and ψ its parametrization. As seen in Definition 5.1.2, f (x, y) = y · g(x, y) with mt(g) = 2, i(g, y) = 4 and δ(f ) = q + 5. By Proposition 1.2.14 that q + 5 = δ(f ) = 0 + δ(g) + i(y, g).

5.2 A disjoint list of IUS

117

and then δ(g) = q + 1 since i(g, y) = 4. By Lemma 1.2.18, there exists a coordinate change Φ ∈ AutK K[[x, y]] in the form Φ(x) = x, and Φ(y) = y + c1 xd1 + c2 xd2 + . . . , 2 = d1 < d2 < . . . . such that 2 - k, where k = ord Φ(g)(0, y). Then as shown in the proof of Lemma 2.5.4, δ(Φ(g)) = k−1 . Thus k = 2q + 3 since δ(Φ(g)) = δ(g). 2 Moreover, it follows from Proposition 1.2.21 that a parametrization of Φ(g) is of the form (at2 + O(3)|bt2q+3 + O(2q + 4)). Hence the parametrization ψ ◦ Φ of Φ(f ) has the form t c1 t2 + O(3) . at2 + O(3) bt2q+3 + O(2q + 4) This implies that Φ(f ) and hence R is of type T2q+3,3,2 . The inclusion T2q+3,3,2 ⊂ E2,2q−1 follows from reading the above proof argument backward. (ii) This is done by the same method as in (i). (iii)-(v) follow easily from Remark 1.2.3 and simple calculations.

The following table (Table 5.2) obtains from Table 5.1 by excluding E2,q and Z0,q . The above proposition shows that a plane curve singularity is ideal-unimodal if and only if it is of type E, T, W or Z in Table 5.2. To distinguish the ideal-unimodal singularities we need to consider more invariants of plane curve singularities. Lr Let R be a plane curve singularity, m its maximal ideal and R ,→ R0 = i=1 K[[t]] its normalization. For each system of generators {x, y} of m with x = (x1 , . . . , xr ), y = (y1 , . . . , yr ) ∈ R ⊂ R0 =

r M

K[[t]]

i=1

we denote



 tord(x1 ) tord(y1 )   .. .. 2 LM(x, y) :=   ∈ R0 . . . tord(xr ) tord(yr )

Let C be the conductor ideal of R0 in R, then C is a principal R0 -ideal of form h(tc1 , . . . , tcr )i. We recall that c(f ) := (c1 , . . . , cr ) the conductor of f and that mt(f ) := (mt(f1 ), . . . , mt(fr )) and c(f ) := (c(f1 ), . . . , c(fr )) the multi-multiplicity and multi-conductor of f respectively, with c(fi ) the conductor of i-th branch fi . Proposition 5.2.2. The conductor (c), multi-multiplicity (mt) and the multiconductor (c) of the IUS’s are given as in the following table (Table 5.3) For the proof we need the following lemmas, which is a simple consequence of

118


Type

r

v(x)

v(y)

E

1 2

(3) (1, 2)

(l) (∞, l)

T

W

Z

3 (1, 1, 1) 2 (2, k) 3 (1, 1, k) 4 (1, ∞, 1, k) 1 (4) 2 (1, 3) 2 (2, 2)

(∞, 3, k) (l, 2) (∞, l, 2) (∞, 1, l, 1) (l) (∞, l) (3, 3)

3 2 3 4

(∞, l, 3) (∞, 3) (∞, 1, l) (∞, 1, 2, k)

(1, 1, 2) (1, l) (1, ∞, 2) (1, ∞, 1, 1)

Condition

Name

l = 7, 8, 10, 11 l = 4, 5, l = 6, 7 k≥3 k ≥ l odd, kl > 4 k odd, lk ≥ 2 k≥l≥1 l = 5, 6, 7 (∗) l = 4, 5 (∗) l ≥ 5, odd l≥2 l = 4, 5, 7, 8 l = 3, 4, 5 k≥2

E2l−2 E2,2q−1 , E13 E3,2q−1 , E19 E3,2(k−1) Tk+2,l+2,2 Tk+2,2l+2,2 T2k+2,2l+2,2 ] W12 , W1,2q−1 , W18 W13 , W17 ] W1,0 , W1,2q W1,l−3 W1,2l−3 Z2l+3 Z12 , Z1,2q−1 , Z18 Z1,2(k−1)

Table 5.2: The index q depends on the delta-invariant of the corresponding singularity. More precisely, δ(E2,2q−1 ) = q + 5 and δ(E3,2q−1 ) = q + 8. ] ] δ(W1,0 ) = 8, δ(W1,2q−1 ) = q + 7 and δ(W1,2q ) = q + 8. δ(Z0,2q−1 ) = q + 5, δ(Z1,2q−1 ) = q + 8. (∗) If char(K) = 2, extra conditions in case W are required: if r = 1 then l = 5, if r = 2, v(x) = (2, 2) and l = 3 then x2 − ay 3 6∈ t7 R0 for all a ∈ K. ] (This excludes W18 (r = 1) and all W1,q (r = 1 or r = 2).) Lemma 5.2.3. If f is an irreducible plane curve singularity having a parametrization (x(t), y(t)) with ordx(t) = n, ordy(t) = m and gcd(m, n) = 1. Then 2δ(f ) = (m − 1)(n − 1). Proof. Follows immediately from Proposition 2.2.10. Lemma 5.2.4. Let f1 , f2 be irreducible with parametrization (x1 (t), y1 (t)) and (x2 (t), y2 (t)) respectively. Assume that m1 n2 6= m2 n1 , where ni = ordxi (t) and mi = ordyi (t). Then i(f1 , f2 ) = min{m1 n2 , m2 n1 }. Proof. Follows immediately from Proposition 2.2.10. Proof of Proposition 5.2.2. The proof for IUS’s without index q, follow easily from Proposition 1.2.15 and Lemmas 5.2.3, 5.2.4. For the IUS’s indexed by q we prove ] ] only for E3,2q−1 since the proofs for W1,2q−1 , W1,2q , Z1,2q−1 work along the same lines. Assume that f ∈ E2,2q−1 : Then f = f1 ·f2 with mt(f1 ) = 1 and mt(f2 ) = 2. Note that δ(f ) = q + 8, and by Lemma 5.2.4, i(f1 , f2 ) = 6. It follows from Proposition

5.2 A disjoint list of IUS

Name

r

E2l−2

1

E13 , E19

2

E3,2q−1

2

E3,2(k−1)

3

Tk+2,l+2,2

2

Tk+2,2l+2,2

3

T2k+2,2l+2,2

4

W12 , W18 ] W1,2q−1

1 1

W13 , W17

2

W1,0

2

] W1,2q

2

W1,l−3

2

W1,2l−3

3

Z2l+3

2

Z12 , Z18

3

Z1,2q−1

3

Z1,2(k−1)

4

LM(x, y) 3 t t 2 t t 2 t  t  t t 2 t tk  t  t k t t  0   t tk t4 4 t t 3 t 2 t 2 t2 t 2 t2 t 2  t t  t 2 t t l t  t  0 2  t t  0 2 t t  0   t t

tl 0 tl 0 t6  0 t3  tk tl t2  0 tl  t2  0 t   tl  t tl t6 0 tl t3 t3 t3 t3 t3 tl  0 tl  t3 0 t3  0 t  tl  0 t  t4  0 t   t2  tk

119

Condition l = 7, 8, 10, 11 l = 5, 7

k≥3 ( k ≥ l, odd kl > 4 ( k odd lk ≥ 2

k≥l≥1 l = 5, 7 (∗) (∗) l = 4, 5

(∗) l ≥ 5, odd l≥2 l = 4, 5, 7, 8 l = 3, 5

k≥2

mt 3 1 2 1  2  1  1  1 2 2   1  1   2  1  1     1  1 4 4 1 3 2 2 2 2 2  2  1  1  2 1  3  1  1   2  1  1   2  1  1     1  1

c

c

2l − 2 2l − 2 0 l l − 1 2l − 1 0 6 2q + 4 2q + 10    0 k+3  0   6  0 k+3 l−1 l+3 k−1 k+3     0 l+2  0   l+2  k − 1  k+3  0 l+2  0   k+2       0   l+2  0 k+2 3l − 3 3l − 3 2q + 14 2q + 14 0 l 2l − 2 3l − 2 2 8 2 8 2 q+8 2 q+8 2 8 l − 1 l + 5     0 l+3  0   l+3  2 8 0 3 2l − 2 2l    +1  0 l+1  0   3   l − 1   2l + 1  0 5  0   3  2q +  2   2q + 8  0 k+3  0   3       0   5  0 k+3

Table 5.3: r: number of branches; mt(f ): multi-multiplicity; c(f ): multi-conductor; c: conductor.

120


1.2.14 that δ(f2 ) = q + 2 and hence c(f )1 = 2δ(f1 ) + i(f1 , f2 ) = 6; c(f )2 = 2δ(f2 ) + i(f1 , f2 ) = 2q + 10 due to Proposition 1.2.15. Proposition 5.2.5. All classes in Table 5.2 are mutually disjoint. Proof. Note that if f is parametrization equivalent to g, then r(f ) = r(g). Combining Proposition 1.2.19 and Corollary 1.2.20 we get c(f ) = c(g), mt(f ) = mt(g), and c(f ) = c(g) (up to a permutation of the indices {1, . . . , r}). Hence the claim follows from Proposition 5.2.2.

5.3

Pre-normal form

The conditions in Table 5.1 characterize the parametrization of the corresponding singularities to a certain extent. In Table 5.4 below we write down the parametrization explicitly where we have made already some simplifications with the help of coordinate changes and reparametrization, getting a kind of “pre-normal form”. In the following, for each m ≤ n, we denote by t[m,n] an element of K-vector space generated by {tm , tm+1 , . . . , tn } and set t[m] := t[m,m] = {tm }. Hence t[m] = a · tm for some a ∈ K. Proposition 5.3.1. Tables 5.4 and 5.5 are a full list of parametrization of idealunimodal singularities. ] ] Proof. We prove only for singularities of types E3,2q−1 , W1,2q−1 , W1,2q (note that the ] ] singularities of type W1,2q−1 , W1,2q are defined only in the case char(K) 6= 2). The proofs for E3,2q−1 , Z0,2q−1 , Z1,2q−1 work analogously. The proof of the remaining IUS follow easily from Propositions 1.2.6, 5.2.2 and Theorem 2.5.3. Case 1: Assume that f is of type E3,2q−1 . It follows by the same argument as in proof of Proposition 5.2.1(i) that f is contact equivalent to a singularity having a parametrization in the following form c1 t3 + O(4) t . ψ := at2 + O(3) bt2q+5 + O(2q + 6)

We see that 0

ψ ∼ ψ :=

t t3 + O(4) t2 + O(3) t2q+5 + O(2q + 6)

,

by the coordinate change x 7→ αx, y 7→ βy and by the reparametrization t t/α 7→ , t t/u where u is a (2q − 1)-th root of b/(a3 c1 ); α = u2 /a and β = u2q+5 /b.

5.3 Pre-normal form

121

Table 5.4: Pre-normal form of parametrization of IUS in char 6= 2 Type E

T

W

r

Normal form

[l+1,2l−3] 1 t3 + t[4,2l−3] tl + t t 0 2 2 l [l+1,2l−2] t t3 + t [4,5] t t +t 2 2 2q+5 [2q+6,2q+9]  t t + t t t 3  t t3  t atk 2 t tl + t[l+1,l+2] 2 tk + t[k+1,k+2] t2   t 0 tl + t[l+1]  3  t k [k+1] t2  t +t  t 0  0  t  4   t tl + t[l+1]  atk + t[k+1] t 1 t4 tl + t[l+1,3l−4] 1 t4 t6 + at2q+5 + t[2q+6,2q+13] t 0 2 3 [4,3l−3] t + t tl+ t[l+1,3l−3] 2 3 t t 2 3 2 + t[4,7] t2 at 3 t t 2 2 3 q+3 [q+4,q+7] + t t2 t3 + at t t 2 2 l [l+1,l+4] t t +t  t 0 3  t tl + t[l+1,l+2]  t2 t3 + t[4,7]

Condition

Name

l = 7, 8, 10, 11

E2(l−1)

l = 5, 7

E13 , E19

q≥1 ( k ≥ 3, a 6= 1 ( k ≥ l odd, lk > 4 ( k odd, lk ≥ 2

E3,2q−1 E3,2(k−1) Tk+2,l+2,2 Tk+2,2l+2,2

( k ≥ l ≥ 1, a 6= 1

T2k+2,2l+2,2

l = 5, 7 q ≥ 1, a 6= 0

W12 , W18 ] W1,2q−1

l = 4, 5

W13 , W17

a 6= 1

W1,0

q ≥ 1, a 6= 0

] W1,2q

l ≥ 5, l odd

W1,l−3

l≥2

W1,2l−3

122


Type

r

Normal form

Condition

0 t l = 4, 5, 7, 8 l [l+1,2l] t3 +t[4,2l] t +t t 0  0 t  l = 3, 5 2 l [l+1,2l] t t + t   t t2 + t[3,4]  0 t  q≥1 2q+3 [2q+4,2q+7] 2 +t   t t t 0 (   0 t k ≥ 2,   2 [3,4]   t t +t a= 6 1 k [k+1,k+2] t t +t

Name

Z

2 3

3

4

Z2l+3 Z12 , Z18

Z1,2q−1

Z1,2(k−1)

• If char(K) 6= 2, Proposition 1.2.6 (or Lemma 4.2.4) yields that there exists a reparametrization φ such that φ ◦ ψ 0 has the following form t t3 + O(4) , t2 t2q+5 + O(2q + 6) which is parametrization equivalent to t t3 + t[4,5] t2 t2q+5 + t[2q+6,2q+9] by Theorem 2.5.3, since the conductor of a E3,2q−1 -singularity is

6 2q + 10

.

• If char(K) = 2, it follows from Lemma 4.2.4 and Theorem 2.5.3 that ψ 0 is equivalent to t t3 + t[4,5] t2 + t[3,2q+9] t2q+5 Conversely if f = f1 · f2 is a plane curve singularity having parametrization in the form ψ1 (t) t t3 + O(4) = , ψ2 (t) t2 + O(3) t2q+5 + O(2q + 6) where ψi (t) is a parametrization of fi , i = 1, 2. Since f1 is regular, it follows from Lemma 1.2.18 that there exists a coordinate change Φ ∈ AutK K[[x, y]] such that Φ(f1 ) = y. Then it is easy to see that Φ(f ) has a parametrization of form t 0 , t2 + O(3) t6 + O(7) which is actually of type E3,2q−1 . ] Case 2: Assume that f ∈ W1,2q−1 (char(K) 6= 2).

5.3 Pre-normal form

123

Table 5.5: Pre-normal form of parametrization of IUS in char 2 Type E

T

W

r

Normal form

1 t3 tl + t[l+1,2l−3] t t2 + t[3] 2 2 [3,2q+5] t2q+3 t +t t 0 2 2 [3,2l−2] tl t +t t t3 + t[4,5] 2 t2 + t[3,2q+9] t2q+5   t t 3  t tl  t atk 2 t + t[3,l+2] tl 2 tk t2 + t[3,k+2]   t 0 3  t tl + t[l+1]  2 [3,k+2] k  t t +t  t 0  0  t  4  l [l+1]   t t +t atk + t[k+1] t [l+1,3l−4] 4 1 t + t[5,3l−4] tl + t t 0 2 3 l [l+1,3l−3] t2 t + t t t3 2 2 + t[3,7] t3 at t2 t3 2 2 [3,l+4] tl t +t  t 0 tl + t[l+1,l+2]  3  t 2 [3,7] t +t t3

Condition

Name

l = 7, 8, 10, 11 E2(l−1) q≥1

E2,2q−1

l = 5, 7

E13 , E19

q≥1   l = 2, 3, k ≥ l,   a 6= 1 ( l, k odd lk > 4 ( k odd, lk ≥ 2

E3,2q−1 El,2(k−1)

Tk+2,l+2,2 Tk+2,2l+2,2

( lk ≥ 1, a 6= 1

T2k+2,2l+2,2

l = 5, 7

W12 , W18

l = 4, 5

W13 , W17

a 6= 1

W1,0

l ≥ 5, odd

W1,l−3

l≥2

W1,2l−3

124

Type


r

Normal form

Condition

Name

0 t l [l+1,2l+1] t3 t +t  0 t  0  t 2 [k] l [l+1,2l] t +t  t +t t + t[2,3] t  0 t 2 [k] t2q+1 + t[2q+2,4q+2] t + t  t t2 + t[3,4]  0 t 2 [k] t + t t2q+3 + t[2q+4,2q+7]   t 0  0 t    l [l+1,2l]  t t +t  k [k+1,k+l] t at + t

l = 4, 5, 7, 8 ( l = 2, 3; k odd k ≤l+1 ( q ≥ 1, k odd k ≤ 2q + 1 ( q ≥ 1, k odd k ≤ 2q + 3   l = 1, 2, k ≥ l,   a 6= 1

Z2l+3

Z

2 3

3

3

4

   

Z12 , Z18

Z0,2q−1

Z1,2q−1

Zl−1,2(k−1)

Since the valuation type of f is (4, 6), by Proposition 1.2.6, we may choose a parametrization of f in the form (t4 |a6 t6 + a7 t7 + . . .), a6 6= 0. Put k := min{i ≥ 7|ai 6= 0, i odd}. We shall show that ψ ∼p (t4 |a6 t6 + ak tk + O(k + 1)). Put ψ0 := ψ and i0 := inf{i > 6|ai 6= 0, i even}. If i0 > k then the claim is obvious. Assume that i0 < k. Since gcd(4, 6) = 2 and since i > 6 even, there exist u0 , v0 ≥ 0, (u0 , v0 ) 6= (0, 1) such that 4u0 + 6v0 = i0 . Then it is easy to see that the coordinate change x 7→ x, y 7→ y − ai0 /av60 xu0 y v0 transforms ψ0 into ψ1 : ψ1 = (t4 |b6 t6 + bi1 ti1 + . . . + bk tk + O(k + 1)), where i0 < i1 even, b6 = a6 , bk = ak , k = min{i ≥ 7|bi 6= 0, i odd}. We continue in this fashion obtaining ψ = ψ0 ∼p ψ1 ∼p . . . ∼p ψn = (t4 |a6 t6 + ak tk + O(k + 1))). It follows from Proposition 2.3.9 that k = 2q + 5 since δ(ψn ) = δ(f ) = q + 7. ] Case 3: Assume that f ∈ W1,2q (char(K) 6= 2). Then f = f1 · f2 with mt(f1 ) = mt(f2 ) = 2 and δ(f1 ) = δ(f2 ) = 1 due to Proposition 5.2.2. It follows from Corollary 2.5.7 that f1 is contact equivalent to g1 = x3 − y 2 . We write f ∼c g = g1 · g2 . Note that i(g1 , g2 ) = q + 6 since δ(g) = δ(f ) = q + 8 and δ(g1 ) = δ(g2 ) = 1. Let (t2 , y(t)) be the Puiseux parametrization of g2 . Then q + 6 = i(g1 , g2 ) = ordg1 (t2 , y(t)) = ord(t6 − y(t)2 ).

5.3 Pre-normal form

125

This implies that y(t) is of form y(t) = t3 + atq+3 + O(q + 4) with a 6= 0. Hence the claim follows from Theorem 2.5.3.

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Wissenschaftlicher Werdegang

Name: Nguyen Hong Duc Geburtsdatum: Dezember 27, 1982 Geburtsort: Nghe An, Vietnam 2000: Arbitur am Phan Dang Luu Gymnasium, Nghe An, Vietnam 2004: Bachelor in Mathematik an der “Hanoi National University of Education” 2006: Master in Mathematik, Institut f¨ ur Mathematk Hanoi ab 2009: Promotion an der University of Kaiserslautern

Scientific Career

Name: Nguyen Hong Duc Date of birth: December 27, 1982 Place of birth: Nghe An, Vietnam 2000: Graduation (Arbitur) at Phan Dang Luu High School, Nghe An, Vietnam 2004: Bachelor in Mathematics, Hanoi National University of Education 2006: Master in Mathematics, Hanoi Institute of Mathematics from 2009: Promotion at the University of Kaiserslautern