Some Results on Secant Varieties Leading to a Geometric Flip

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induced by cubics vanishing on the secant variety, and the exceptional locus should ... Theorem E is proved in Theorem 5.3.4 (where a natural extension of this theorem for smooth ... dimension; thus answering a question of A. Bertram [B3]. ..... of the closure of the graph Γϕ for any pair (X, Fi), not necessarily satisfying (Kd).
Some Results on Secant Varieties Leading to a Geometric Flip Construction by Peter Vermeire

A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics.

Chapel Hill 2005

Approved by

Advisor: Professor Jonathan Wahl

Reader: Professor James Damon

Reader: Professor Philippe DiFrancesco

Reader: Professor Shrawan Kumar

Reader: Professor Michael Schlessinger

ABSTRACT PETER VERMEIRE: Some Results on Secant Varieties Leading to a Geometric Flip Construction (Under the direction of Jonathan Wahl) We study the relationship between the equations defining a projective variety and properties of its secant varieties. In particular, we use information about the syzygies among the defining equations to derive smoothness and normality statements about SecX and also to obtain information about linear systems on the blow up of projective space along a variety X. We use these results to geometrically construct, for varieties of arbitrary dimension, flips first described in the case of curves by M. Thaddeus via Geometric Invariant Theory. We conclude by examining the problem of cubic generation of secant varieties and by giving a construction of further flips under similar, though not yet well-understood, hypotheses.

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CONTENTS Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1. Condition (Kd ) on a Projective Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2. Secant and Tangent Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3. Secant Bundles on Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3. The Structure of a Map ϕ on the Complement of a Variety . . . . . . . . . . . . . . . . 11 3.1. The Resolution of ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2. The Fibers of ϕ fn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3. Properties of |kH − mE| on P g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 e Restricted to SecX 3.4. ϕ 3.5. Work of A. Bertram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.6. Deficiency of Secant Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4. Completing the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1. The General Birational Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2. The Exceptional Loci. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3. The Exceptional Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4. The Total Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.5. Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5. Further Flips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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5.1. Work of M. Thaddeus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2. Cubic Generation of Secant Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3. Construction of the Second Flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.4. Construction of Further Flips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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CHAPTER 1

Introduction Let X ⊂ Pn be a variety, scheme theoretically defined by quadrics F0 , . . . , Fs . The Fi define a rational map ϕ : Pn 99K Ps z 7→ [F0 (z), . . . , Fs (z)] which is defined wherever the Fi are not all zero, i.e. on Pn \ X. The first portion of this paper is devoted to the study of the map ϕ and its resolution ϕ e : BlX (Pn ) → Ps . 1 Note the following elementary observation: If L ∼ = P is a secant line to X, L 6⊂ X, then the restriction of ϕ to L is a linear system of quadrics on L with two base points. As two points on P1 determine a unique quadric, this system must be zero dimensional; i.e. ϕ will collapse L to a point in Ps . The union of the set of such lines forms the secant variety to X, denoted SecX. Hence we see that ϕ restricted to SecX will collapse secant lines to points. We can ask two natural questions: Is ϕ an embedding away from SecX ⊂ Pn ? And does ϕ separate secant lines? That is, are distinct secant lines always mapped to distinct points in Ps ? The first question clearly fails whenever X is defined by fewer than n+1 equations. The second fails whenever a P2 intersects X in four distinct points, as then there is a pair of secant lines which intersect away from X, and so must be mapped to the same point in Ps . Hence, some further hypotheses are needed. Many familiar projective varieties are defined ideal theoretically by quadrics. For example: rational normal curves, Veronese embeddings of Pn , Pl¨ ucker embeddings of Grassmannians, and Segre embeddings of Pn × Pm . A more informative and nontrivial example is any smooth curve of genus g embedded by a line bundle of degree d ≥ 2g + 2. In fact, if X is an arbitrary (not necessarily smooth) variety and L is an ample line bundle, then the image of X under the embedding given by L ⊗k is ideal theoretically defined by quadrics for all k  0. Recall that a syzygy among the equations {Fi } is a vector of homogeneous forms (a0 , . . . , as ) such that formally s X ai Fi = 0 i=0

By a theorem of M. Green [Gr], a smooth curve X embedded by a line bundle of degree ≥ 2g + 3 is ideal theoretically defined by quadrics, and the module of syzygies among the quadrics is generated by linear syzygies, (i.e. where the ai have degree 1). In fact, the above examples (including sufficiently large powers of ample line bundles)

have this property which, coupled with projective normality, is usually referred to as Green’s property (N2 ). With the exception of some easy to describe cases, if X satisfies property (N2 ) then the answer to both questions posed above is yes. In fact a slightly weaker condition (K2 ) (Definition 2.1.2) suffices. Theorem A. Let (X, Fi ) be a pair that satisfies (K2 ), and assume that X does ^ the proper transform of SecX. not contain a line. Then ϕ e is an embedding off SecX, Furthermore, if X is smooth then the image of ϕ e is a normal subvariety of Ps . Theorem B. Let (X, Fi ) be a pair satisfying (K2 ), and assume X ⊂ Pn is smooth, irreducible, contains no lines, and contains no plane quadrics. Then the restriction ^ is a P1 -bundle over Hilb2 (X). Hence SecX ^ is smooth, SecX is smooth of ϕ e to SecX off X, and SecX is normal. Theorem A is given in more general form in Theorem 3.2.4. Theorem B is a combination of Theorem 3.4.6 with Corollary 3.4.7, where an explicit rank 2 vector bundle on Hilb2 (X) is identified. fn → Ps , can be explicitly The key to proving these results is that the fibers of ϕ e:P described in terms of the linear syzygies among the Fi . In particular, every fiber is given by the vanishing of a collection of linear forms. Chapter 4 is motivated by the observation that if codim(SecX, Pn ) ≥ 2, then ϕ e is a small morphism of projective varieties. This is simply the statement that ϕ e is an isomorphism in codimension one. By analogy with work of A. Bertram [B],[B3] and M. Thaddeus [T] outlined in Chapter 5, one expects that there is a flip diagram associated to ϕ. e The first example of a flip was given by M. Atiyah [A] by constructing two nonisomorphic partial resolutions of an isolated 3-fold singularity (the affine cone over a smooth quadric hypersurface in P3 ). Strictly speaking, this is an example of a flop, but this difference will not be pursued here. The theory of flips has gained prominence in recent years in Mori’s theory of Minimal Models (or the Minimal Model Program) Cf. [Ko],[M2]. For us, a flip means simply a birational surgery in codimension at least two which changes the ample cone. Specifically, we construct a smooth variety fn in codimension one, hence there is an isomorphism M2 which is isomorphic to P fn ). By a change in the ample cone we mean there are line bundles in Pic(M2 ) ∼ = Pic(P fn ) which are not ample on P fn , but whose images in Pic(M2 ) are ample on M2 . Pic(P If X is a smooth curve embedded by a line bundle of degree at least 2g + 3, then the space M2 exists as a GIT quotient by work of M. Thaddeus [T]. Our construction proceeds as follows: Blow up the proper transform of the secant fn (which is shown to be smooth in Corollary 3.4.7). By identifying ^⊂P variety SecX natural vector bundles E , F on Hilb2 (X), we construct a diagram of exceptional loci (Diagram 4.3.1) as a diagram of projective bundles of various dimension. In the case of smooth curves, these bundles are identified by Thaddeus. The picture is

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uu π uuu u uu u zu

P(E )

JJ JJ JJ J ϕ e JJ$

E2 I II II h2 II II I$ P(F ) t tt tt t t ty t f

Hilb2 X

~ π ~~ ~ ~ ~~ ~

fn P BB

f2 M A

AA h2 AA AA A

{ BB {{ { BB B {{  ϕe B }{{ f Pn _ _ _/ Ps

M2

ϕ

fn f2 = Bl where M ^ (P ). What remains is to construct a variety M2 containing P(F ) SecX f2 → M2 whose restriction is the map h : E2 → P(F ). and an explicit morphism h2 : M By studying linear systems on these projective bundles, we identify a base point f2 whose restriction to the exceptional locus is the map given free linear system on M in the aforementioned diagram. Taking M2 to be the image of this linear system, the flip diagram is completed. Theorem C. Let (X, Fi ) satisfy (K2 ) and assume X ⊂ Pn is smooth, irreducible, contains no lines and contains no plane quadrics. Then there is a flip as pictured above with: fn , M f2 , and M2 smooth (1) P (2) P(E ) and P(F ) are projective bundles over Hilb2 X fn \ P(E ) ∼ (3) P = M2 \ P(F ) (4) h2 , induced by OM ((2k − 1)H − kE1 − E2 ) for k sufficiently large, is the g 2 blowing up of M2 along P(F ) fn along P(E ) (5) π is the blowing up of P (6) f , induced by OM2 (2H − E), is an isomorphism off of P(F ), and the restriction of f is the projection P(F ) → Hilb2 X (7) ϕ, e induced by OPfn (2H−E), is an isomorphism off of P(E ), and the restriction of ϕ e is the projection P(E ) → Hilb2 X This is Theorem 4.5.5. Smoothness of M2 is proven in Proposition 4.5.3. In Chapter 5, we discuss extensions of these ideas to further flips. By the work of Thaddeus, one expects a birational morphism ϕ e2 : M2 → Ps2 whose exceptional 2 3 locus is a P -bundle over Hilb (X). By analogy with our earlier work, ϕ e should be induced by cubics vanishing on the secant variety, and the exceptional locus should be the transform of Sec2 X, the variety of 3-secant 2-planes to X. The first obstacle is to find when SecX is scheme theoretically defined by cubics. This is known to be the case in some special examples (Cf. Remark 5.2.1), but there are no general theorems in the style of quadric generation discussed above. We prove (Theorem 5.2.2): Theorem D. Let X ⊂ Pn satisfy condition (K2 ). Then Sec(v2 (X)) is set theoretically defined by cubics. Taking this as evidence that large embeddings of a projective variety should have cubically defined secant variety, we prove:

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Theorem E. Let X ⊂ Pn be a smooth, irreducible curve embedded by a line bundle of degree at least 2g + 5. Assume that the following conditions are satisfied: (1) Sec1 X is scheme theoretically defined by cubics whose trivial syzygies are generated by linear ones (2) Sec2 X is scheme theoretically defined by forms of degree at most 5 Then the morphism ϕ e2 : M2 → Ps2 induced by OM2 (3H − 2E) is an isomorphism off of the transform of Sec2 X, and ϕ e2 restricted to the transform of Sec2 X is a P2 -bundle over Hilb3 (X). Furthermore, there exists a second flip as in Theorem C. Theorem E is proved in Theorem 5.3.4 (where a natural extension of this theorem for smooth varieties of arbitrary dimension is given) and the discussion that follows it. In Section 5.4, a general procedure for constructing further flips is outlined. However, it should be noted that it relies on some (at present) fairly difficult facts about secant varieties. Motivation for this work includes: • Understanding the very natural map ϕ, see [CK1],[CK2],[ES-B],[GL],[HKS]. • Deducing properties of the secant variety from conditions on the syzygies among the equations defining a projective variety. • Obtaining a natural, geometric way to construct those flips resulting from the work of M. Thaddeus. In particular, to construct Thaddeus’ flips without the machinery of Geometric Invariant Theory, and to extend the construction to arbitrary dimension; thus answering a question of A. Bertram [B3]. • Exploiting the birational geometry of the blow up of projective space along a smooth variety X to obtain sharp theorems on the cohomology of IXr (r + s) (Cf. [BEL],[B4],[T],[W]). The philosophy is that the ample cone on the flipped spaces fn , allowing one to use standard (e.g. M2 above) will be different from that on P fn . For example, one would vanishing theorems whose results could be pulled back to P like to answer a conjecture of J. Wahl [W] related to the extendibility of canonical curves. We will decorate a projective variety X as follows: X d is the dth cartesian product d of X; S d X is Symd X = X /Sd , the dth symmetric product of X; and Hd X is Hilbd (X), the Hilbert Scheme of zero dimensional subschemes of X of length d. Recall (Cf. [Go]) that if X is a smooth projective variety then Hd X is also projective, and is smooth if either dim X ≤ 2 or d ≤ 3. If V is a k-vector space, we denote by P(V ) the space of 1-dimensional quotients of V . Unless otherwise stated, we work throughout over the field k = C of complex numbers. We use the terms locally free sheaf (resp. invertible sheaf) and vector bundle (resp. line bundle) interchangeably. If D ⊂ X is a Cartier divisor, then the associated invertible sheaf is denoted OX (D). We conform to the convention that products of line bundles corresponding to explicit divisors are written additively, while other products are written multiplicatively, e.g. (L ⊗ OX (D))⊗n ∼ = L ⊗n ⊗ OX (nD). A line bundle

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L on X is nef if L .C ≥ 0 for every irreducible curve C ⊂ X. A line bundle L is big if L ⊗n induces a birational map for all n  0. I would like to thank the members of my committee: James Damon, Philippe DiFrancesco, Shrawan Kumar, and Michael Schlessinger for their time and energy. I would also like to thank the following people for their helpful conversations and communications: Aaron Bertram, Lawrence Ein, Stephanie Fitchett, Anthony Geramita, Klaus Hulek, Shreedhar Inamdar, Vassil Kanev, S´andor Kov´acs, Mario Pucci, M. S. Ravi, Michael Schlessinger, Michael Thaddeus, and Jerzy Weyman. I would lastly like to thank my advisor Jonathan Wahl for his guidance, enthusiasm and limitless patience in teaching me the subject of Algebraic Geometry and for suggesting this line of research. Without him this work would not have been possible.

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CHAPTER 2

Preliminaries 2.1. Condition (Kd ) on a Projective Variety Recall that a syzygy among a set of polynomials F0 , . . . , Fs of degree d is a vector r = (a0 , . . . , as ) of forms of degree k such that formally: s X

ai Fi = 0

i=0

If k = 1 this is referred to as a linear syzygy. One always has trivial syzygies: tij = (0, . . . , Fj , . . . , −Fi , . . . , 0) where Fj is in the ith position and −Fi is in the j th position. In [Gr], M. Green defines condition (N2 ) for a projective variety X ⊂ Pn as: (1) X is projectively normal (2) X is ideal theoretically defined by quadrics Fi (3) All of the syzygies among the Fi are generated by linear ones Example 2.1.1 Examples of varieties that satisfy (N2 ) include: (1) X a smooth curve embedded by a line bundle L with deg(L) ≥ 2g + 3 [Gr] (2) If X ⊂ Pg−1 is a canonical curve with Cliff(X) > 2, then X satisfies (N2 ) [Sc],[Vo]. (3) All Veronese embeddings of Pn , e.g. [EL]. See [Laz] for an introduction to these ideas and more examples. 2 We make the following: Definition 2.1.2. Let X be a subscheme of Pn . X satisfies condition (Kd ) if X is scheme theoretically cut out by forms F0 , . . . , Fs of degree d such that the trivial (or Koszul) relations among the Fi are generated by linear syzygies. More generally, let V ⊆ H 0 (OPn (d)) be a linear system of polynomials of degree d with (possibly empty) base scheme X. Then the pair (X, V ) satisfies condition (Kd ) if the trivial (or Koszul) relations among the elements of V are generated by linear syzygies. Write (X, Fi ) for the pair (X, V ) if the set {Fi } generates the linear system V . If X is a hypersurface of degree d, or if V is a zero dimensional linear system, then condition (Kd ) is vacuously satisfied.

Clearly, condition (N2 ) implies (K2 ). Though (K2 ) is a technically simpler condition and arises naturally, in practice most examples we consider that satisfy (K2 ) will actually satisfy the stronger condition (N2 ). Remark 2.1.3 The reason for the two definitions of (Kd ) is as follows: We will use the rational map Pn 99K P(V ) induced by a linear system satisfying condition (Kd ) to deduce results about secant varieties to a projective variety. However, most of these results will depend only on the existence of such a system, and not on the particular map chosen. 2 A linear system satisfying (Kd ) has the following nice property, which is clear from the definition: Lemma 2.1.4. Let V be a linear system on Pn that satisfies (Kd ), and let L ∼ = Pk n be a linear subspace of P . Then V restricted to L satisfies (Kd ). Corollary 2.1.5. Let L be an ample line bundle on a variety X. Then the embedding of X by L⊗r satisfies (K2 ), for all r  0. Proof: Let X ⊂ Pn be a projectively normal embedding induced by some L⊗k and assume that X is scheme theoretically defined by forms of degree d. The image of the d-tic Veronese embedding of Pn satisfies (N2 ), hence (K2 ), and vd (X) is a linear section of this corresponding to the embedding induced by L⊗kd . 2 Remark 2.1.6 Corollary 2.1.5 remains true if (K2 ) is replaced by (N2 ), though the proof of this fact is much harder, especially for X singular [Gr],[In]. 2 Lemma 2.1.7. Let (X, Fi ) satisfy (Kd ), X ⊂ Pn . Then (X, {zj Fi |∀i, j}) satisfies (Kd+1 ), where Pn = Proj(C[z0 , . . . , zn ]). Proof: Begin by noticing that a syzygy among F0 , . . . , Fs , say: (a`0 , . . . , a`s ) gives an induced syzygy among: z0 F0 , z0 F1 , . . . z0 Fs , z1 F0 , . . . , zn Fs simply by concatenation: (a`0 , . . . , a`s , a`0 , . . . , a`s , . . . , a`s ) Denote this induced syzygy by (a`0 , . . . , a`s )0 .

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A trivial syzygy among the zj Fi can now be written as follows: (0, . . . , zα Fj , . . . , −zβ Fi , . . . , 0) = (0, . . . , zα Fj , . . . , −zα Fi , . . . , 0) +(0, . . . , zα Fi , . . . , −zβ Fi , . . . , 0) X = zα b` (a`0 , . . . , a`s )0 `

+Fi (0, . . . , zα , . . . , −zβ , . . . , 0) where the (a`0 , . . . , a`s ) are the linear syzygies that generate the trivial syzygy between Fi and Fj , and zα Fi is put in the (β, j) position. 2 Recall the following definition: Definition 2.1.8. Let X ⊂ P(W ) be a smooth variety, W ⊆ H 0 (X, L) for some line bundle L. W is k-very ample if the map W ⊗ OX → H 0 (L ⊗ OZ ) induced by restriction is surjective for every subscheme Z ⊂ X of length ≤ k. Hence W is 1-very ample if it is base point free and 2-very ample if it is very ample. Note that this implies that every length k subscheme of X determines a (k − 1)plane in P(W ). In the case of a smooth curve X, a line bundle L is k-very ample if and only if 0 h (X, L(−p1 − p2 − . . . − pk ))+k = h0 (X, L) for any pi ∈ X, not necessarily distinct. Example 2.1.9 Suppose that X ⊂ P(W ) is scheme theoretically defined by quadrics and does not contain a line, where W is a sub vector space of H 0 (X, L) for some line bundle L. Then since any quadric vanishing 3 times on a line must vanish on the entire line, X cannot have a trisecant line. Hence W is 3-very ample. 2 Proposition 2.1.10. Let X ⊂ P(W ) = Pn , n ≥ 3 be a smooth, irreducible variety that satisfies condition (K2 ), contains no lines, and contains no plane quadrics. Then W is 4-very ample. Proof: Assume to the contrary that there is a 2-plane H that intersects X in a scheme Z of length k, where k is greater than 3 (note that by hypothesis H cannot intersect X in a scheme of positive dimension). A linear system of quadrics on P2 with zero dimensional base scheme Z has length Z = 4 if and only if the linear system consists of two quadrics [Hart, V.4.2]. However, if there are two, they cannot satisfy (K2 ) unless they share a linear factor, which would imply that Z has positive dimension. Therefore, there can be no such 2-plane. 2 The quadratic Veronese embeddings of Pn , for example, require the inclusion of the hypothesis that X contains no plane quadrics.

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2.2. Secant and Tangent Varieties We define here carefully what we mean when we refer to secant and tangent varieties to a projective variety X ⊂ Pn (see [H, Chapters 8,15] for further details and discussion). Let X ⊆ Pn be a non-degenerate projective variety, let G(`, n) be the Grassmannian of `-planes in Pn , and let X k denote the k th cartesian product of X. For k < n, define the rational map: sk : X k+1 99K G(k, n) by sending a (k + 1)-tuple of points to the k-plane they span. Note that if k = 1, this map will be defined off of the diagonal ∆, while for k ≥ 2, the indeterminacy locus will in general be larger than the union of the partial diagonals. A simple argument shows that sk is defined on an open set. Resolve this map via the closure of the graph: Γsk I II II π2 II π1 II I$  sk k+1 _ _ _/ G(k, n) X and obtain a subvariety Sk (X) = Im π2 of G(k, n). Looking at the incidence correspondence: Σ ⊂ G(k, n) × Pn π10



NNN 0 NNNπ2 NNN NNN N'

G(k, n)

Pn

we define Seck X = π20 ((π10 )−1 (Sk (X))), the k th secant variety to X in Pn . In the case k = 1, we similarly define T anX, the tangent variety to X in Pn , by  looking at the subvariety T (X) = π2 π1−1 (∆) of G(1, n). Note the following result: Theorem 2.2.1. ([FL]) If X is irreducible (but not necessarily smooth) of dimension k, then one of the following holds: (1) dim Sec1 X = 2k + 1 and dim T anX = 2k (2) Sec1 X = T anX It will be useful to understand the image of the map sk : Proposition 2.2.2. Let X be a projective variety, and suppose the embedding of X ,→ Pn is (k +2)-very ample. Then the closure of the image of sk : X k+1 99K G(k, n) is a reduced component of Hilbk+1 (X) ⊂ G(k, n). Proof: It is shown in [CG] that the natural map Hilbk+1 (X) → G(k, n) is an embedding under the hypothesis stated. 2

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2.3. Secant Bundles on Projective Varieties We describe vector bundles on Hk+1 (X) = Hilbk+1 (X) and morphisms to projective space giving rise to Seck X. Our construction follows that of [B, §1], [ACGH, VIII.2], and [S] where this is done for curves with the identification of Hk+1 (X) with S k+1 (X). All technical statements made here concerning Hilbert Schemes can be found in [Go], and all schemes are taken over Spec(C). Define the universal subscheme D ⊂ X × Hk+1 X, which is flat of degree k + 1 over Hk+1 X, by the following universal property: For every locally noetherian scheme U and every subscheme Z ⊂ X × U which is flat of degree k + 1 over U there is a unique morphism: f : U → Hk+1 X such that: Z = (idX × f )−1 (D) As a set, D = {(p, D) ∈ X × Hk+1 X|p ∈ Supp(D)}. Note that for k = 1, D ∼ = Bl∆ (X × X). Let π : X × Hk+1 X → X and πk+1 : X × Hk+1 X → Hk+1 X be the projections, and let L be any line bundle on X. Form the invertible sheaf π ∗ L on X × Hk+1 X, and, by restriction, the invertible sheaf OD ⊗ π ∗ L on D ⊂ X × Hk+1 X. Now πk+1 |D : D → Hk+1 X is flat of degree k + 1. Therefore, EL = (πk+1 )∗ (OD ⊗ π ∗ L) is a locally free sheaf of rank k + 1 on Hk+1 X. We define the k th secant bundle of X with respect to L to be B k (L) = PHk+1 X (EL ). To define the desired map, look at the natural restriction map of sheaves on X × Hk+1 X: π ∗ L → OD ⊗ π ∗ L Pushing this down to Hk+1 X gives an evaluation map: H 0 (X, L) ⊗ OHk+1 X → EL which in turn restricts, for any linear system V ⊆ H 0 (X, L), to a map: V ⊗ OHk+1 X → EL Now, a fiber of EL over a point Z ∈ Hk+1 X is H 0 (X, L ⊗ OZ ). So if V is (k + 1)-very ample then this map is surjective, and we obtain a morphism: B k (L) → P(V ) × Hk+1 X → P(V ) Let U ⊂ Hk+1 X be the open subscheme parameterizing smooth subschemes of length k + 1, U its closure in Hk+1 X. Then the image of the above morphism restricted to BUk (L) ≡ PU (EL ) is the k th secant variety to X in P(V ). Note that U need not equal Hk+1 X (Cf. [Ia]). Note that the surjection V ⊗ OHk+1 X → EL → 0 induces the obvious morphism Hk+1 X → G(k, V ∗ ) mentioned in the proof of Proposition 2.2.2.

10

CHAPTER 3

The Structure of a Map ϕ on the Complement of a Variety 3.1. The Resolution of ϕ Let V ⊆ H 0 (Pn , OPn (d)) be a linear system on Pn with base scheme X. There is then a rational map ϕ : Pn 99K P (V ) = Ps . Let Γϕ ⊂ Pn × Ps be the closure of the graph of the rational map ϕ. ϕ is then naturally resolved: Γϕ G GG GG ϕe GG π1 GG G#  ϕ X ⊂ Pn _ _ _/ Ps where π1 is the restriction of the projection onto the first factor and ϕ e is the restriction of the projection onto the second factor. Lemma 3.1.1. π1 : Γϕ → Pn is the blow up of Pn along X. Proof: Let IX be the ideal sheaf of X. Because X is scheme theoretically defined by forms of degree d, there is a surjection of sheaves: OPn (−d)s+1

//I X

from which we get a surjection: k L

Si (OPn (−d)s+1 )

//

i=1

k L

(IX )i

i=1

for all k, and so taking Proj, a closed embedding: BlX (Pn ) ,→ Pn × Ps Now both BlX (Pn ) and Γϕ are complete, reduced and irreducible subvarieties of P × Ps , and the set-theoretic image of BlX (Pn ) coincides with Γϕ off of X in Pn . Therefore, BlX (Pn ) and Γϕ are isomorphic as schemes. 2 n

Remark 3.1.2 If a linear system V is not specified, we take V = Γ(Pn , IX (d)). The value of d will be clear from the context.

e 3.2. The Fibers of ϕ If M ∼ = P1 is a d-secant line to X, then the points of intersection determine a unique d-tic on M , hence the restriction of ϕ to M collapses M to a point. Similarly, we see that ϕ e will collapse to a point the proper transform of any linear subspace of Pn that intersects X in a d-tic hypersurface. What we show is that the fiber over any point of the image of ϕ e is the proper transform of such a linear subspace of Pn , or is a reduced point. In order to better understand ϕ e it is helpful to obtain a more explicit description of the closure of the graph Γϕ for any pair (X, Fi ), not necessarily satisfying (Kd ). Lemma 3.2.1. Let (X, Fi ) be any pair and let [z0 , . . . , zn ; t0 , . . . , ts ] be coordinates for Pn ×Ps . Let S be the subscheme of Pn ×Ps defined by the bihomogeneous equations {Fi (z)tj − Fj (z)ti = 0} of bidegree (d, 1). Then Γϕ ⊆ S, and S = Γϕ , as schemes, off of E, the exceptional divisor of the blow up. Proof: Every equation Fi (z)tj − Fj (z)ti vanishes on Γϕ , hence Γϕ ⊆ S as schemes. Now, choose a point s = [p0 , . . . , pn ; q0 , . . . , qs ] ∈ S. Say qi 6= 0 for some i and, because we exclude the exceptional divisor, assume that Fj (p) 6= 0 for a particular j. We then have: Fi (p)qj − Fj (p)qi = 0 which implies that qj 6= 0. Therefore, for any k: qk Fk (p) = qj Fj (p)

(1)

which says that s = [p0 , . . . , pn ; F0 (p), . . . , Fs (p)], and so S = Γϕ off of E. Note that if i = j above, we move directly to (1). 2 Note that S = Γϕ if and only if X is a scheme theoretic complete intersection defined by the Fi , or is empty. s X Observe that a homogeneous syzygy among the Fi , say ai (z)Fi (z) = 0, gives a bihomogeneous equation

s X

i=0

ai (z)ti = 0 for Γϕ . If one assumes that the pair (X, Fi )

i=0

satisfies condition (Kd ), then the trivial relations are generated by linear syzygies. Let {(a`0 , . . . , a`s ); 0 ≤ ` ≤ r} generate the linear syzygies among the Fi . Writing the trivial syzygies in terms of these linear syzygies, define the closed set T ⊂ Pn × Ps by the bihomogeneous equations ( s ) X a`k (z)tk ; 0 ≤ ` ≤ r k=0

12

(Cf. [HKS, §1]). It is easy to verify that Γϕ ⊆ T ⊆ S as schemes. Proposition 3.2.2. Let (X, Fi ) satisfy (Kd ) and assume that X does not contain a line. If a ∈ Im ϕ, e then ϕ e−1 (a) = Pk × {a} ⊂ Pn × Ps , where either (1) k = 0 or (2) π1 (Pk × {a}) = Pk ⊂ Pn intersects X in a hypersurface of degree d in Pk . Proof: By Lemma 3.2.1, Γϕ = T off of E, the exceptional locus of the blow-up. ϕ e−1 (a) is contained as a scheme in Ta , where Ta is the fiber over a of the projection map restricted to T . Without loss of generality, apply a change of coordinates to the space V so that a = [1, 0, . . . , 0] ∈ Ps . Ta is then scheme theoretically defined by the bihomogeneous equations: ) ( s s X X a0k tk , . . . , ark tk , t1 , t2 , . . . , ts k=0

k=0

and so is more simply defined by: {a00 t0 , . . . , ar0 t0 , t1 , t2 , . . . , ts } giving: Ta = Pk × [1, 0, . . . , 0] ⊂ Pn × Ps ∼ = Pk where Pk is the linear subspace of Pn defined by the {a`0 }. From the beginning of the proof, ϕ e−1 (a) = Ta off of E. Ta is irreducible, however, so ϕ e−1 (a) = Ta ∼ = Pk as long as either of the following is true: (1) Ta is a reduced point. (2) Ta and ϕ e−1 (a) are not both contained in E. To guarantee that the second possibility occurs if Ta has positive dimension, note that X does not contain a line. Therefore π1 (Ta ), which is a linear subspace of Pn , cannot be isomorphic to a positive-dimensional reduced linear subscheme of X. Hence Ta cannot be contained in E. 2 Remark 3.2.3 The proof shows the Proposition holds if X contains a line, but is not set theoretically cut out by any subset of the Fi . Also, it holds if one requires only that the varieties V (Ji ) are not positive dimensional linear subspaces of X, where Ji is the ideal generated by linear forms r such that rFi ∈ (F0 , . . . , Fbi , . . . , Fs ). Note that in the proof of the above Proposition, Ta ∼ 2 = V (J0 ). As an application of Proposition 3.2.2 we have:

13

Theorem 3.2.4. Let (X, Fi ) be a pair that satisfies (Kd ), and assume that X does not contain a line. Then ϕ e is an embedding off of the proper transform of Sec1d (X), the variety of d-secant lines. Furthermore, if X is smooth then the image of ϕ e is a s normal subvariety of P . Proof: To prove the first claim, note that because the fibers of ϕ e are reduced, we need only show that points are separated. Take p, q ∈ Γϕ , and assume ϕ(p) e = ϕ(q) e = {r} ∈ Ps . Then by Proposition 3.2.2, −1 k S=ϕ e (r) satisfies π(S) = P , k > 0. There are then two possibilities: (1) π(S) ∩ X is a d-tic hypersurface in π(S), and hence every line in π(S) is a d-secant line of X, which implies that π(S) ⊆ Sec1d (X), and so p and q are in the proper transform of the variety of d-secant lines. (2) π(S) ⊆ X. In this case, however, π(S) is a positive dimensional linear subvariety of X, which is not allowed by hypothesis. To see that the image is normal if X is smooth, notice that Γϕ is smooth, hence normal, because by Lemma 3.1.1 it is the blow up of a smooth variety along a smooth subvariety. We have just shown that the fibers are reduced and connected, and so we are done by the following Lemma 3.2.5. 2 Lemma 3.2.5. Let f : X → Y be a proper surjective morphism of irreducible varieties over an algebraically closed field k with reduced, connected fibers. If X is normal then Y is normal. Proof: Because X is normal, f factors uniquely through the normalization Ye of Y : fe

/ e Y ?? ?? g ? f ?? 

X?

Y where g is a finite, hence proper, morphism. The properness of f and g implies that fe is proper ([Hart, II.4.8.e]), hence its image is closed in Ye . Surjectivity of f implies that dim Im(fe) = dim Y = dim Ye , and so fe is surjective. Because f has reduced, connected fibers, g does also, and so as g is finite the fibers of g are reduced points. Now for any point p ∈ Y , look at the local ring A with e be its normalization. Tensoring the exact sequence: maximal ideal mp and let A A

g∗

/ e A

e /A

e mp A

e /A

/A

/0

by ⊗A A/mp gives: k

e/ /A

/A ⊗A k

14

/0

e/ e ∼ k. e is maximal, hence A Because the fibers of g are reduced points, mp A mp A = e/ ⊗ k = 0, which implies A e/ = 0 by Nakayama’s Lemma and so g is Therefore, A A A A an isomorphism. 2 Corollary 3.2.6. Let X ⊂ Pn be a smooth, irreducible, non-degenerate curve that satisfies Green’s condition (N2 ), and let {Fi } be the quadrics that vanish on X. Then ϕ e is an embedding off of the proper transform of Sec1 X. Proof: By the hypotheses, X cannot contain a line and we can apply Theorem 3.2.4. 2 Remark 3.2.7 We do not know if the exclusion of the case where X contains a line in Theorem 3.2.4 is necessary. 2 Remark 3.2.8 The proof of Theorem 3.2.4 implies that if X ⊂ Pn is scheme theoretically defined by forms of degree d that satisfy (Kd ), then the map ϕ : Pn \ X → Ps is an embedding off of Sec1d (X), even if X does contain a line. This result (for d = 2) was discovered independently by K. Hulek and W. Oxbury [HO]. 2 Example 3.2.9 The quadratic Veronese X = v2 (P2 ) ⊂ P5 is ideal theoretically defined by the rank 1 minors of the matrix:   x0 x1 x2  x1 x3 x4  x2 x4 x5 The cubic equation defining the secant variety is given by the determinant. X satisfies (N2 ), hence the map ϕ : P5 99K P5 is birational by Theorem 3.2.4. The image of the secant variety is again the quadratic Veronese surface in P5 . To see why this is reasonable, note first that any two points on X lie on a plane quadric in P5 (the image under v2 of the line joining them in P2 ), hence every secant line to X lies on a P2 spanned by a plane quadric. The map ϕ e will collapse these planes to points, hence there is an identification of lines in P2 with points in the image of the secant variety. However, the space of lines in P2 is (P2 )∗ . Another way to look at this is to note that, as the secant variety to X is deficient, SecX = T anX (Theorem 2.2.1). It can be checked that the image of the secant variety is its dual variety in (P5 )∗ , which is isomorphic to the quadratic Veronese embedding of P2 . Similarly, the dual variety to X is Sec(X ∗ ) ⊂ (P5 )∗ , hence this map can be thought of as turning P5 “inside out”, swapping X and SecX. 2

15

Example 3.2.10 Let X ⊂ P4 be the twisted quartic. X is cut out by six quadrics which satisfy (N2 ), hence there is a birational map ϕ : P4 99K P5 . It can be verified directly that the image of the secant variety is again the quadratic Veronese surface. 2 Example 3.2.11 Let X be two reduced points in P2 , and look at the system of quadrics vanishing on X, e.g. H 0 (P2 , IX (2)) = (x2 , xy, xz, yz). The quadrics give a map ϕ : P2 99K P3 whose image is a smooth quadric hypersurface: the Segre embedding of P1 × P1 ⊂ P3 . It is very easy to see why this is so, as ϕ is resolved by blowing up the two points of X, and then ϕ e collapses only the one secant line between them. 2 In summary: If X is smooth and satisfies (K2 ) and if Sec1 X is a proper subvariety of Pn then the image of ϕ e in Ps is a rational, normal variety, smooth off of the image of the proper transform of the secant variety, with homogeneous coordinate ring isomorphic to C[F0 , . . . , Fs ] ⊂ C[x0 , . . . , xn ]. fn 3.3. Properties of |kH − mE| on P fn = BlX (Pn ), then If X ⊂ Pn is smooth and irreducible, and if we denote P fn = ZH + ZE where H is the proper transform of the generic hyperplane and E Pic P is the exceptional divisor. Theorem 3.2.4 gives information about the linear systems |kH − mE|. Proposition 3.3.1. Let X ⊂ Pn be smooth, irreducible and satisfy (Kd ). (1) If Sec1d (X) is a proper subvariety of Pn , then |dH − E| is base point free and defines a birational morphism. In particular, it is big and nef. (2) If X does not contain a line, then |αH − E| is very ample if α > d, α ∈ Z. Proof: The first statement follows directly from the hypothesis on Sec1d (X) and Remark 3.2.8. The second statement follows from Theorem 3.2.4 and the fact that if (X, Fi ) satisfies (Kd ) then (X, {zj Fi |∀i, j}) satisfies (Kd+1 ) (Lemma 2.1.7), and so 1 |(d + 1)H − E| gives an embedding off of Sec^ (X). However, as X is scheme d+1

theoretically defined by forms of degree d, Sec1d+1 (X) is empty.

2

Corollary 3.3.2. Let X ⊂ Pn be smooth and irreducible, contain no lines, and satisfy (Kd ). If Sec1d (X) is non-empty, then: (1) |kH − mE|, k, m ∈ Z, is very ample if and only if (2) |αH − E|, α ∈ Q, is ample if and only if α > d

16

k m

> d, m 6= 0

Proof: Let k ≥ md + 1, then O(kH − mE) = O((m − 1)(dH − E)) ⊗ O((k + d − md)H − E) where the first factor is globally generated and the second is very ample by Proposition 3.3.1. The necessity of k > md follows from the existence of a d-secant line. The second statement follows directly from the first. 2 Proposition 3.3.3. Let (X, V ) satisfy (Kd ). Let X ⊂ Pn be smooth and irreducible of codimension e. If Sec1d (X) is a proper subvariety of Pn , then: H i (Pn , IXa (k)) = 0, i > 0, k ≥ d(e + a − 1) − (n + 1) Proof: fn be the blow-up of Pn along X. Since Let P KPfn = OPfn ((−n − 1)H + (e − 1)E) we have for i ≥ 0:   fn , O fn (kH − aE) (Cf. [BEL]) H i (Pn , IXa (k)) = H i P P   fn , K fn ⊗ O fn ((n + k + 1)H − (e + a − 1)E) = Hi P P P By Proposition 3.3.1, |(n + k + 1)H − (e + a − 1)E| is big and nef as long as: n+k+1 ≥ d e+a−1 k ≥ d(e + a − 1) − (n + 1) The result follows by the Kawamata-Viehweg vanishing theorem [Ka], [V]. 2 Remark 3.3.4 Proposition 3.3.3 is the natural extension of the result (Cf. [BEL]) that if X ⊂ Pn is a smooth, irreducible variety of codimension e that is scheme theoretically defined by forms of degree d, then: H i (Pn , IXa (k)) = 0, i > 0, k ≥ d(e + a − 1) − n The improvement here is due simply to the fact that if one assumes (Kd ), then OPfn (2H − E) is big.

17

^ e Restricted to SecX 3.4. ϕ In this section, X ⊂ Pn is a smooth, irreducible, non-degenerate variety, scheme theoretically defined by quadrics F0 , . . . , Fs satisfying (K2 ). Assume that X contains no lines and no plane quadric curves. This assumption will be crucial to the discussion below. In this situation, Theorem 3.2.4 implies that the map ϕ e is an embedding off the proper transform of the secant variety to X. Here we study what the map does when restricted to the proper transform of the secant variety. Our main results are ^ is isomorphic to a Theorem 3.4.6 and its Corollary 3.4.7, where we show that SecX 1 2 P -bundle over H X, extending a result of A. Bertram [B] for curves. ^ the proper transform of the secant Write SecX for Sec1 X and denote by SecX n f → Pn of Pn along X. By a slight abuse of variety under the blowing up π : P fn → Ps . ^ → Ps for the restriction of ϕ notation, write ϕ e : SecX e:P ^ → H2 X, and then an embedding of H2 X We show first that there is a map SecX ^ → Ps . into the image variety of ϕ, e such that the composition factors ϕ e : SecX Remark 3.4.1 Because of the assumption that X contains no lines and no plane quadrics, each ^ → Ps is isomorphic to P1 by Proposition 3.2.2. In particular, given fiber of ϕ e : SecX ^ or in SecX \ X, one can say exactly which secant or tangent line it a point in SecX lies on. 2 There is a diagram: H2 X u:

g u u u u ϕ SecX _ _ _ _/ Ps

where g, defined as a set map on SecX \ X, takes a point p to the length 2 subscheme Z of X determining the secant line on which p lies. This map is well-defined by Remark 3.4.1. Blowing up SecX along X: g e ^ _ _ _/ H2 X SecX H

HH ϕe HH HH HH $

Ps

Lemma 3.4.2. ge extends to a morphism. Proof: We do this indirectly by construct a morphism to G(1, n) whose image is H2 X. Denote Y = Im ϕ, e and push the surjection ^ O ^ (H)) ⊗ O ^ → O ^ (H) → 0 H 0 (SecX, SecX SecX SecX

18

down to Y : ^ O ^ (H)) ⊗ OY → ϕ e∗ OSecX H 0 (SecX, ^ (H) SecX The sheaf ϕ e∗ OSecX ^ (H) is locally free of rank 2, and the map is surjective as O(H) maps a fiber of ϕ e to a linearly embedded P1 ⊂ Pn . Pulling this surjection back ^ gives a surjection from a free rank n + 1 sheaf to a rank 2 vector bundle, to SecX ^ → G(1, n) taking a fiber of ϕ hence a morphism SecX e to the point representing the associated secant line. The image of this morphism is clearly H2 X ,→ G(1, n) from Proposition 2.2.2. 2 ^ → H2 X, we construct an embedding f : Having constructed a map ge : SecX ^ → Ps . As above, we first describe f as a set H2 X ,→ Ps so that ϕ e = (f ◦ ge) : SecX map. Let Z ∈ H2 X be a length 2 subscheme of X, and let `Z ⊂ Pn be the line determined by Z. Note that by hypothesis `Z does not lie on X. There are homomorphisms: rZ : V → H 0 (Pn , OPn (2) ⊗ O`Z ) Associate to every Z ∈ H2 X the 1-dimensional quotient V /ker(rZ ). This gives a set map f : H2 X → P(V ) = Ps which is a set-theoretic injection by Remark 3.4.1. Lemma 3.4.3. f : H2 X → P(V ) = Ps is a morphism. Proof: We construct on H2 X a surjection from a trivial rank s+1 = dim V vector bundle to a line bundle, inducing a morphism which agrees with f as a set map. fn × H2 X. Embed: Let L = OPfn (2H) and form π1∗ L on P fn × H2 X ^ ,→ P SecX p 7→ (i(p), ge(p)) fn is the inclusion. ^ ,→ P where i : SecX Applying π2∗ to the surjection: π1∗ L → π1∗ L ⊗ OSecX ^ → 0 gives a map: fn , L) ⊗ OH2 X → π2∗ π ∗ L ⊗ O H 0 (P 1 ^ SecX



fn , L ⊗ O fn (−E)) gives a map (by composition): Letting V = H 0 (P P  V ⊗ OH2 X → π2∗ π1∗ L ⊗ OSecX ^  where a fiber of the coherent sheaf π2∗ π1∗ L ⊗ OSecX over a point Z ∈ H2 X is ^ 0 n isomorphic to H (P , OPn (2) ⊗ O`Z ). By the above remarks, this map has rank 1, hence gives a surjection to a line bundle on H2 X with fiber over Z isomorphic to H 0 (Pn , OPn (2) ⊗ IX ⊗ O`Z ). f is the morphism induced by this surjection. 2

19

The diagram g e

/ H2 X HH ϕe HH f HH HH  $

^ SecX H

Ps then commutes, and because the fibers of ϕ e are reduced, those of f are as well: Proposition 3.4.4. Let (X, V ) be a pair satisfying (K2 ), and assume X ⊂ Pn is smooth, irreducible, contains no lines, and contains no plane quadrics. Then the morphism f : H2 X ,→ P(V ) above is an embedding. 2 Lemma 3.4.5. With hypotheses as in Proposition 3.4.4, the exceptional divisor ^ → SecX is isomorphic to Bl∆ (X × X). of the blow up SecX Proof: ^ be the exceptional divisor of this blow up. Let Y be the image of: Let F ⊂ SecX F → X × H2 X p 7→ (π(p), ϕ(p)) e Y is then flat of degree 2 over H2 X, and by the representability of the Hilbert functor induces the identity morphism id : H2 X → H2 X. By the universal property of D ∼ = Bl∆ (X × X), the universal subscheme of X × H2 X, we have: Y ∼ = (idX × idH2 X )−1 (D) The map from F to Y is then a finite birational morphism to a smooth variety, and so is an isomorphism, hence F ∼ = Bl∆ (X × X). 2 fn ∼ This allows another construction of the secant bundle B 1 (L): Writing Pic P = ^ The ZH + ZE, form the line bundle H ⊗ OF on the exceptional divisor F ⊂ SecX. 2 restriction of ϕ e to F is a degree two flat map to H X. Define E =ϕ e∗ (H ⊗ OF ) By the identification of F with D and of H ∼ = π ∗ OPn (1) with L on X, B 1 (L) ∼ = PH2 X (E ). We come to the main result of this section: Theorem 3.4.6. Let X ⊂ Pn be smooth, irreducible and satisfy (K2 ). If X ^∼ ^ → H2 X contains no lines and no plane quadrics then SecX e : SecX = B 1 (L), i.e. ϕ is a P1 -bundle. Proof: Note that ϕ e∗ (H ⊗ OF ) and ϕ e∗ (H ⊗ OSecX ^ ) are isomorphic rank two vector bundles 2 on H X (this follows immediately from the fact that H restricted to a fiber of ϕ e is 20

OP1 (1), and the fact that H 0 (P1 , O(−1)) = H 1 (P1 , O(−1)) = 0). This and the fact that H ⊗ OSecX ^ is generated by its global sections implies there exists a surjection ϕ e∗ E → H ⊗ OSecX ^ → 0 which induces a morphism ^ → B 1 (L) κ : SecX This gives the diagrams: / B 1 (L) u uu π uu u u  zuu β1

/ B 1 (L) II ϕe II p II II $ 

κ

κ

^ SecX

^ SecX I

SecX H2 X where p is the natural projection map. κ makes both triangles commute, and so is a finite (by the second diagram) birational (by the first) morphism to a smooth variety, hence an isomorphism. 2 Corollary 3.4.7. Under the hypotheses of Theorem 3.4.6: ^ is smooth and SecX is smooth off of X (1) SecX (2) T^ anX is smooth and T anX is smooth off of X (3) SecX is normal Proof: ^ is immediate from Theorem 3.4.6. The smoothness of SecX To see that the proper transform of the tangent variety is smooth, note that ϕ e 2 maps T^ anX to the diagonal in H X, which is the projectivized tangent bundle to X, hence smooth (by the diagonal in H2 X, we mean the proper transform on the diagonal under the birational morphism H2 X → S 2 X). Therefore, T^ anX is smooth as above. ^ is smooth To show SecX is normal, we use the (just proven) fact that SecX −1 and Lemma 3.2.5. Hence it suffices to show that for p ∈ X, π (p) is reduced and connected where ^ → SecX π : SecX is the blow up of SecX along X. But by Lemma 3.4.5, π −1 (p) ∼ = Blp (X) 2 3.5. Work of A. Bertram ^ is isomorphic to B 1 (L) if In [B], A. Bertram is able to show directly that SecX X is a smooth curve embedded by the complete linear system associated to a line bundle L that is 4-very ample. We outline his proof here.

21

Begin by looking at the map β1 : B 1 (L) → P (H 0 (X, L)) = PL . One shows first that β1−1 (X) is isomorphic to X × X by looking at the map: α : X × X → B 1 (L) defined by sending   (p, q) 7→ P H 0 (X, L/L(−p))   which is in P H 0 (X, L/L(−p − q)) , the fiber over [p, q] ∈ S 2 X. One then sees that this map is an embedding and checks that, as a set, α(X×X) = β1−1 (X). One can then reduce equality of schemes to showing that the restriction map H 0 (X, L(−2p)) → H 0 (X, L(−2p) ⊗ Oq ) is always surjective, which is true as long as L is 3-very ample. fL . This map is clearly an embedding This now allows us to lift β1 to βe1 : B 1 (L) → P off of α(X × X), and it takes p × X into the fiber P (H 0 (X, L(−2p))) over the point p ∈ X. Via a diagram chase involving the the normal and tangent sheaves of the relevant objects, Bertram is able to show that βe1 is in fact an embedding. 3.6. Deficiency of Secant Varieties A simple consequence of the above results is that a smooth, irreducible variety X ⊂ Pn satisfying (K2 ) with no lines and no plane quadrics has a non-deficient secant variety. In this section, we drop the requirements that X have no lines and no plane quadrics and study the deficiency of the secant variety. Proposition 3.6.1. Let X be an n-dimensional variety satisfying (K2 ), and let ^ → Ps be the map defined earlier with image variety Y . Let δ = 2n + 1 − ϕ e : SecX dim(SecX) be the deficiency of the secant variety to X. Then: 2n − dimY 2 In particular, the dimension of Y is always even. δ=

Proof: Let r be the dimension of the general fiber of ϕ, e so that dim(SecX) = r + dimY . Now we know that the general fibers of ϕ are linear subspaces of dimension r intersecting X in quadric hypersurfaces. Therefore, the general point of SecX lies on an (r − 1)-dimensional family of secant lines to X and so: dim(SecX) r + dimY −dimY 2n − dimY 2n − dimY

= = = = =

2n + 1 − (r − 1) 2n + 2 − r 2n + 2 − 2r − 2dimY 4n + 2 − 2dim(SecX) 2δ 2

22

Corollary 3.6.2. If X is a variety satisfying (K2 ), then SecX is deficient if and only if the general pair of points of X lie on a plane quadric. Remark 3.6.3 Note that our definition of deficiency here is a bit different than usual, as the standard definition presupposes that SecX is a proper subvariety of projective space. 2 Example 3.6.4 Corollary 3.6.2 allows us to see quickly that the only Veronese embeddings of Pn with deficient secant varieties are the quadratic embeddings (and, in fact, that these have δ = 1). We also see that if X ⊂ Pn is non-degenerate and defined by forms of degree d, then vr (X), the image of X under the r-tic Veronese embedding, has a non-deficient secant variety if r ≥ d. This is a special case of [L, 10.3], where this is shown to be true for r ≥ 2, i.e. independent of the forms defining X. 2

23

CHAPTER 4

Completing the Diagram In this chapter, motivated by M. Thaddeus [T], we construct a flip centered about H2 X. In Sections 4.2 and 4.3 we construct a diagram of the exceptional loci (Diagram 4.3.1), in Section 4.4 we construct the actual flip (Diagram 4.4.8), and finally in Section 4.5 we investigate the smoothness of the spaces constructed. Our results are summarized in the main result of this chapter, Theorem 4.5.5. With notation as above assume X ⊂ Pn satisfies (K2 ), contains no lines, and contains no plane quadrics. Let r = dim X and assume that n − 2r − 1 ≥ 2, i.e. that SecX is not a hypersurface in Pn . Write P(E ) for the secant bundle PH2 X (E ) ^ with P(E ) (Theorem 3.4.6). We abuse notation and simply write and identify SecX fn → Ps . ϕ e : P(E ) → H2 X for the restriction of ϕ e:P To help the reader understand the rather long construction about to be described, we include a short section giving a general overview of the construction along with an explicit description of what will be shown in the remainder of the chapter. 4.1. The General Birational Construction Let f : X 99K Y and g : X 99K Z be rational maps of irreducible varieties, f birational. Form Γf,g ⊂ X × Y × Z, the closure of the graph of (f, g) : X 99K Y × Z, and let M2 ⊂ Y × Z be the image of Γf,g under the obvious projection. We have then the following diagrams where all maps are projections, and the left included in the right by restriction: ~~ ~~ ~ ~ ~~ ~

Γf,g

Γf B B

BB BB BB !

Y

AA AA AA AA

{ {{ {{ { { }{ {

M2

X × Y ×L Z

LLL LLL LLL LL&

r rrr r r rrr rx rr

X × YN NNN NNN NNN NNN &

Y ×Z

Y

qq qqq q q qqq qx qq

Note that X, Y, Γf , Γf,g , and M2 are all birational. Example 4.1.1 A variation of the above proceeds as follows: Let f : X 99K Y be a rational map of projective varieties and assume that Γf → Y is a small morphism with exceptional locus W ⊂ Γf . We then have an embedding BlW (Γf ) ⊂ X × Y × Ps for some s, where the blow up BlW (Γf ) → Γf is given by restricting the natural

projection. Projecting X × Y × Ps → Y × Ps then gives a divisorial contraction BlW (Γf ) → M2 , hence Γf and M2 are isomorphic in codimension 1. 2 In this chapter, we give a more explicit construction of this birational transformation. In particular, building on the work done in the previous chapters, we take fn → Ps . As the exceptional locus of ϕ ^ we form Γf → Y above to be ϕ e:P e is SecX, fn f2 = Bl M ^ (P ), and the diagram can be completed by projection. We identify SecX explicitly the exceptional loci in this diagram, give explicit linear systems defining the morphisms and finally study the space M2 . Though our construction follows Example 4.1.1, it will turn out in the end that there is a rational map g which would have allowed the use of the first construction to derive the same results. 4.2. The Exceptional Loci We are led by Thaddeus [T, 3.11] to construct a vector bundle F on H2 X of rank fn ) such that: n − 2r − 1 = codim(P(E ), P (2) ϕ e∗ (F ) ∼ ⊗ OP(E ) (−1) = N∗ n f P(E )/P

One of the exceptional loci will be constructed as PH2 X (F ). ∗ Write NP(E ) (k) = NP(E )/Pfn ⊗ OP(E ) (k). We verify F = ϕ e∗ NP(E ) (−1) satisfies (2). Recall that a vector bundle E on a projective variety X is said to be ample, if the line bundle OPX (E) (1) on PX (E) is ample. We record an elementary fact and prove a general statement: Lemma 4.2.1. Let E be a vector bundle on P1 . The following conditions are equivalent: (1) E is L ample ∼ (2) E = OP1 (ai ), ai ≥ 1 (3) H 0 (P1 , E ∗ ) = H 1 (P1 , E ⊗ OP1 (−2)) = 0 (4) E ⊗ OP1 (−1) is generated by global sections 2 Proposition 4.2.2. Let π : P → X be a P1 -bundle, X (and hence P ) smooth. Let E be a vector bundle on P such that the restriction of E to every fiber is trivial. Then π∗ E is locally free and π ∗ π∗ E ∼ = E. Proof: Let Yp = π −1 (p) ∼ = P1 . Riemann-Roch implies immediately that h0 (Yp , EYp ) = rk E, ∀p ∈ X hence π∗ E is locally free of rank rk E [Hart, III.12.9]. Because EYp is globally generated, the natural map π ∗ π∗ E → E is surjective (Cf. [Hart, III.8.8]), hence an isomorphism. 2

25

Proposition 4.2.3. Let Ye ∼ e Then NP(E ) (k) ⊗ OYe ∼ = P1 ⊂ P(E ) be a fiber of ϕ. = ⊕OYe (k − 1). Hence by Proposition 4.2.2, ∗ ∼ ∗ ϕ e∗ ϕ e∗ NP(E ) (−1) = NP(E ) (−1) Proof: We show first that NP(E ) (k) ⊗ OYe is ample for k ≥ 2: Consider  /f Ye Pn π







/ Pn Y where Y is a secant line to X, and hence a linearly embedded P1 , and π is the blow up of Pn along X. Denoting tangent sheaves by T , we have an exact sequence:

0 → Z → TPfn → π ∗ TPn → 0 where Z is supported on E, the exceptional divisor of the blow-up. Tensoring with OYe , we have H i (Z ⊗ OYe ) = 0, i > 0 because Z ⊗ OYe is supported on a zerodimensional scheme. Hence: H 1 (Ye , T fn ⊗ O e ) = H 1 (Ye , π ∗ TPn ⊗ O e ) Y

P

Y

= H 1 (Y, TPn ⊗ OY ) Now, the usual presentation of the tangent bundle on Pn implies that TPn ⊗ OY (2) is ample on Y , hence H 1 (Y, TPn ⊗ OY ) = 0. This implies that TPfn ⊗ OYe (2) is ample on Ye , and so by twisting the sequence: 0 → TP(E ) ⊗ OYe → TPfn ⊗ OYe → NP(E ) ⊗ OYe → 0 we see that NP(E ) (k) ⊗ OYe is ample for k ≥ 2 [Hart2, III.1.7] because it is the image of an ample bundle. By Lemma 4.2.1, we have NP(E ) ⊗OYe ∼ = ⊕OYe (ai ) where the ai ≥ −1. We show via computation of the determinant that ai = −1 and the statement of the Proposition follows. Restrict to Ye the isomorphism of sheaves [Hart, II.8.20]: ωP(E ) ∼ = ω fn ⊗ Λn−2r−1 NP(E ) P

fn = ZH + ZE, we have: Writing Pic P ω fn ∼ = O fn ((−n − 1)H + (n − r − 1)E) P

P

Because X is defined by quadrics, ωPfn ⊗ OYe ∼ = OP1 (n − 2r − 3). Because P(E ) is ∼ ∼ locally a product, ωP(E ) ⊗ OYe = ωYe = OP1 (−2). This gives: Λn−2r−1 NP(E ) ⊗ O e ∼ = OP1 (−n + 2r + 1) Y

2

26

4.3. The Exceptional Diagram In this section, we construct the diagram of exceptional loci for the flip. Let  ∗ E20 = PP(E ) NP(E e∗ F ) ) (−1) = PP(E ) (ϕ and ∗ E2 = PP(E ) NP(E )



fn ). As E2 and E 0 differ Hence E2 is the exceptional divisor of the blow up BlP(E ) (P 2 by the twist of a line bundle, there is an isomorphism γ [Hart, II.7.9]: E2 D D

γ

/ E0 2

DD DD π DD ! 

π0

P(E ) with the property that γ ∗ (OE20 (1)) ∼ = OE2 (1) ⊗ π ∗ (OP(E ) (−1))

(3)

which will be important for bookkeeping purposes. Writing P(F ) = PH2 X (F ), we have the diagram: ww π 0 ww w w w w{ w

P(E )

HH HH HH H ϕ e HH#

E20 H HH 0 HHh HH HH # P(F ) v vv vv v v vz v f

H2 X where f and π 0 are the natural projections and h0 is induced by the natural surjection (ϕ e ◦ π 0 )∗ (F ) → OE20 (1) → 0. Via γ we get a morphism h2 : E2 → P(F ) which, as a morphism of varieties over H2 X, is induced by the surjection (noting (3)):  (ϕ e ◦ π)∗ (F ) → OE2 (1) ⊗ π ∗ OP(E ) (−1) → 0 In summary, we have: Diagram 4.3.1 w ww ww w w w{ w π

P(E )

HH HH HH H ϕ e HH#

E2 H HH HHh2 HH HH $ P(F ) v vv vv v v vz v f

H2 X

27

which will be the diagram of exceptional loci for the flip. The isomorphism γ gives E2 ∼ e∗ F ). Note the following symmetry prop= PP(E ) (ϕ erty: Lemma 4.3.2. E2 ∼ = PP(F ) (f ∗ E ). Proof: To give a map E2 → PP(F ) (f ∗ E ) it is equivalent to give a surjection h∗2 f ∗ E → K → 0 for some line bundle K on E2 . By Diagram 4.3.1, this is equivalent to a surjection π ∗ ϕ e∗ E → K → 0, which we obtain from the natural surjection ϕ e∗ E → OP(E ) (1) → 0 on P(E ). As the fibers of h2 are isomorphic to P1 , it is clear that the induced map is an isomorphism. 2 In the next section, we will use explicit linear systems to construct the flip. It is therefore necessary to embed P(F ) into projective space, and to understand the map h2 as a morphism induced by a complete linear system on E2 . Let the very ample invertible sheaf M on H2 X ⊂ Ps be the restriction of OPs (1). Then for every k sufficiently large, OP(F ) (1) ⊗ f ∗ M k is very ample on P(F ), [Hart, Ex. II.7.14], and so gives an embedding i : P(F ) ,→ Pr . The induced morphism i ◦ h2 : E2 → Pr is given by a linear system associated to the line bundle:  (i ◦ h2 )∗ (OPr (1)) ∼ = h∗2 OP(F ) (1) ⊗ f ∗ M k  ∼ (4) = h∗2 f ∗ M k ⊗ OE2 (1) ⊗ π ∗ OP(E ) (−1) Since h2∗ OE2 = OP(F ) by Lemma 4.3.2, the projection formula yields:   Γ P(F ), OP(F ) (1) ⊗ f ∗ M k = Γ E2 , h∗2 OP(F ) (1) ⊗ f ∗ M k hence: Lemma 4.3.3. The complete linear system |OP(F ) (1)⊗f ∗ M k | on P(F ) pulls back to the complete linear system associated to h∗2 (OP(F ) (1) ⊗ f ∗ M k ) on E2 . 2 4.4. The Total Spaces In this section, we build the total spaces of the flip containing the exceptional loci Diagram 4.3.1, where the maps are given by restriction. Three of the four spaces have fn , Im ϕ, fn ). We construct the fourth (and been constructed already: P e and BlP(E ) (P fn ). This construction most interesting!) as the image of a linear system on BlP(E ) (P fn ) that proceeds in several steps: First, we identify (5) an invertible sheaf on BlP(E ) (P  restricts to h∗2 OP(F ) (1) ⊗ f ∗ M k on E2 (Cf. Lemma 4.3.3). We then show that the associated complete linear system gives a birational morphism, and thatits restriction to E2 is the complete linear system associated to h∗2 OP(F ) (1) ⊗ f ∗ M k . fn ). Writing f2 = BlP(E ) (P Following the notation of Thaddeus [T], denote M f2 = ZH + ZE1 + ZE2 = Z(π ∗ H) + Z(π ∗ E) + ZE2 Pic M

28

and noticing OE2 (−E2 ) = OE2 (1), we have: (5) OE2 ((2k − 1)H − kE1 − E2 ) ∼ = h∗2 f ∗ M k ⊗ OE2 (1) ⊗ π ∗ OP(E ) (−1) Note the similarity with (4). We prove a general statement: Proposition 4.4.1. Let L be an invertible sheaf on a complete variety X, and let B be any locally free sheaf. Assume that for k ≥ N , N ∈ Z, the map fk : X → Yk induced by L k is a birational morphism and that some fk is an isomorphism in a neighborhood of p ∈ X. Then for all n sufficiently large, the map H 0 (X, B ⊗ L n ) → H 0 (X, B ⊗ L n ⊗ Op ) is surjective. Proof: Push the exact sequence 0 → B ⊗ L mk ⊗ Ip → B ⊗ L mk → B ⊗ L mk ⊗ Op → 0 down to Yk . Because fk is an isomorphism in a neighborhood of p, the map fk∗ (B ⊗ L mk ⊗ Op ) → R1 fk∗ (B ⊗ L mk ⊗ Ip ) is the zero map, hence there is an exact sequence: 0 → fk∗ (B ⊗ Ip )(m) → fk∗ (B)(m) → fk∗ (B)(m) ⊗ Op → 0 where L k ' f ∗ OYk (1). Since OYk (1) is (very) ample, H 1 (Yk , fk∗ (B ⊗ Ip )(m)) = 0 for all m sufficiently large. Therefore there is a section of fk∗ (B)(m) that does not vanish at p ∈ Yk which can be pulled back to a section of B ⊗ L km that does not vanish at p ∈ X. 2 Remark 4.4.2 Proposition 4.4.1 should be thought of as an analogue of the statement that if L is an ample line bundle, then L n ⊗ B is globally generated for all n  0. 2 Corollary 4.4.3. For k sufficiently large, the set theoretic base locus of the linear fn is P(E ). system |(2k − 1)H − kE| on P Proof: Clearly, the base locus of |(2k − 1)H − kE| contains P(E ). Note, however, that as |2H − E| is base point free, the base locus of |(2k − 1)H − kE| will stabilize for k sufficiently large. Hence it suffices to show that if p is a point not in P(E ), then |(2k − 1)H − kE| is free at p for all k  0. Now take B = OPfn (−H) and L = OPfn (2H − E) in Proposition 4.4.1, and use Theorem 3.2.4. 2

29

Notation 4.4.4 For the rest of this section, write O(a, b, c) for OM (aH +bE1 +cE2 ), g 2 and write Lk = OM ((2k − 1)H − kE1 − E2 ), k ∈ Q. g 2 Lemma 4.4.5. Lk is nef for k ∈ Z sufficiently large. Proof: f2 be an irreducible curve not contained in E2 , we have Lk .C ≥ 0 Letting C ⊂ M for k  0 by Corollary 4.4.3. Letting C 0 ⊂ E2 , Lk ⊗ OE2 is globally generated on E2 for k  0 by (5), hence Lk .C 0 ≥ 0 and Lk is nef. 2 Proposition 4.4.6. With hypotheses as above and for k sufficiently large, the f2 induced by the linear system |Lk | is an isomorphism off of E2 morphism h2 on M and h2 |E2 = h2 : E2 → P(F ). Proof: We first show that for k sufficiently large |Lk | restricts to the complete linear system on E2 associated to the invertible sheaf h∗2 f ∗ M k ⊗ OE2 (1) ⊗ π ∗ OP(E ) (−1) (Cf. Lemma 4.3.3). For this it suffices to prove:     1 f 1 f = H M2 , O(2k − 1, −k, −2) H M2 , O(2k − 1, −k, −1) ⊗ IE2 = 0 Note that a section of O(2k − 1, −k, −2) is a form on Pn of degree 2k − 1 vanishing k times along X and twice along SecX. Writing B = O(2k − 1, −k, −2) and noting KM = O(−n − 1, n − r − 1, n − 2r − 2): g 2 −1 ∼ B ⊗ KM g = O(2k + n, −k − n + r + 1, −n + 2r) 2

Let α =

k+n−r−1 n−2r

and rewrite the right side as: Lαn−2r ⊗ O(2, 0, 0)

−1 For k  0, Lα is a nef Q-divisor by Lemma 4.4.5, hence B ⊗ KM is big and nef and g 2 the vanishing holds by the Kawamata-Viehweg vanishing theorem (note that by [M, 1.9], a nef line bundle tensored with a big and nef line bundle is again big and nef). To see h2 is a morphism, note that by Corollary 4.4.3, the support of the base scheme is contained in E2 . By what has just been proven, however, h2 has no base ∗ ∗ k points since the  complete linear system on E2 associated to h2 f M ⊗ OE2 (1) ⊗ ∗ π OP(E ) (−1) induces a morphism. Now, possibly twisting by O(2, −1, 0), the rest of the statement holds. 2

Remark 4.4.7 It is unfortunate that this proof gives no bound on k; however, there is an important case when the value of k can be determined. Specifically, if SecX ⊂ Pn is

30

scheme theoretically defined by cubics, then the line bundle L2 will be base point free, hence k = 3 suffices for Proposition 4.4.6. 2 fn \ P(E ) by ProposiDenote by M2 the image variety of h2 . Then M2 \ P(F ) ∼ =P fn and M2 tion 4.4.6. It will be shown (Proposition 4.5.3) that M2 is smooth. Because P fn ∼ are isomorphic in codimension 1, we have [Hart, II.6.5] Pic P = Pic M2 ∼ = ZH + ZE and:   fn , O fn (2H − E) ∼ H0 P = H 0 (M2 , OM2 (2H − E)) P Therefore, OM2 (2H − E) induces a morphism f : M2 → Ps which is an isomorphism off of P(F ). In summary: Diagram 4.4.8 xx π xx x xx {x x

P(E )

GG GG GG G ϕ e GG#

E2 G GG GGh2 GG GG # P(F ) vv vv v vv vz v f

H2 X

~ π ~~ ~ ~ ~~~

fn P BB

f2 M A

AA h2 AA AA A

{{ BB {{ BB { B { }{{ f  ϕe B Pn _ _ _/ Ps

M2

ϕ

fn along P(E ) and the diagram on the left (Diagram 4.3.1) where π is the blowing up of P is given by restriction. Remark 4.4.9 If h2 is the morphism induced by O(5H − 3E1 − E2 ), i.e. if SecX is scheme theoretically defined by cubics, then the ample cone on M2 is spanned by 2H − E and 3H − 2E, i.e. OM2 (aH − bE) is ample if and only if 3 a < 2).

By analogy with our earlier construction, and motivated by [B],[T], we wish to construct a birational morphism ϕ e2 : M2 → Ps2 which contracts the image of 3secant 2-planes to points, and is an isomorphism off of their union. It is easy to verify (Cf. Remark 4.4.7) that the natural candidate for this morphism is the linear system fn ∼ associated to OM2 (3H − 2E), where we identify Pic P = Pic M2 . Noting the fact ∗ that h2 OM2 (3H − 2E) = OM (3H − 2E − E ), it is not difficult to see (using Zariski’s 1 2 g 2 Main Theorem) that this system will be globally generated if SecX ⊂ Pn is scheme theoretically defined by cubics. The picture is as follows: Diagram 5.1.2 ~ ~~ ~ ~ ~~ ~

f2 M A

AA h2 AA AA A

M2

fn P BB

{{ BB ϕe {{ BB ϕ e2 { BB {{ f  ! }  { ϕ Pn S_ _W _/ Ps g k5 Ps2 [ _ c ϕ2

In fact, all of the morphisms can actually be interpreted as projections, much the way ϕ e was earlier. Specifically, if we suppose that SecX ⊂ Pn is scheme theoretically defined by cubics G0 , . . . , Gr then there is a rational map ϕ2 : Pn 99K Ps2 defined by the Gi . This allows us to construct the following diagram: Diagram 5.1.3 Pn × Ps × PPs2

PPP PPP PPP PP(

nn nnn n n nn n w nn

Pn × PsP

Ps × Ps 2

PPP nn PPP nnn n n PPP n PPP nnn   ϕ ( vnnn n W_ _ _ _ _ _ _/ s 3 Ps 2 P P X Y [ f e \ ] ^ _ ` a b c ϕ2

5.2. Cubic Generation of Secant Varieties Remark 5.2.1 Before proceeding any further, it is natural to ask the question: When is the secant variety to a projective variety defined by cubics? Some examples of varieties whose secant varieties are ideal theoretically defined by cubics include: (1) X is any Veronese embedding of Pn [D],[Kan]. (2) X is the Pl¨ ucker embedding of the Grassmannian G(1, n) for any n [H, 9.20]. (3) X is the Segre embedding of Pn × Pm [H, 9.2].

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2 We also prove a general result: Theorem 5.2.2. Let X ⊂ Pn satisfy condition (K2 ). Then Sec(v2 (X)) is set theoretically defined by cubics. Proof: Let Y = v2 (X), V = v2 (Pn ) ⊂ PN , and H the linear subspace of PN defined by the hyperplanes corresponding to all the quadrics in Pn vanishing on X. Then Y = V ∩ H as schemes and we show, noting that SecV is ideal theoretically defined by cubics, that SecY = SecV ∩ H as sets. Note that the map ϕ : Pn 99K Ps can be viewed as the composition of the embedding v2 : Pn ,→ PN with the projection from H, PN 99K Ps . Let p ∈ SecV ∩ H. If p ∈ V , then p ∈ Y = V ∩ H hence p ∈ SecY . Otherwise, any secant line L to V through p intersects V in a length two subscheme Z. Z considered in Pn determines a unique line in Pn whose image in PN is a plane quadric Q ⊂ V spanning a plane M . If H ∩ Q = Z 0 is non-empty then Z 0 ∪ {p} ⊂ H ∩ M , hence either H intersects M in a line L0 through p or M ⊂ H. In the first case L0 is a secant line to Y , in the second Q ⊂ Y . In either situation p ∈ SecY .

All that remains is the case H ∩ M = {p} and hence H ∩ Q is empty. However in this case the line L, and hence the scheme Z ⊂ Q is collapsed to a point by the projection. As the rational map ϕ is an isomorphism off SecX, this implies Z lies on the image of a secant line to X ⊂ Pn . As a length two subscheme of Pn determines a unique line, Q must be the image of a secant line to X ⊂ Pn contradicting the assumption that H ∩ Q is empty. 2 Note: as Green’s (N2 ) implies (K2 ), this shows that the secant varieties to the following varieties are set theoretically defined by cubics: (1) Smooth curves embedded by line bundles of degree 4g +6+2r, r = 0, 1, 2, . . .. (2) Any smooth variety embedded by L⊗2r for all r  0, L ample. (3) Smooth curves X with Cliff X > 2, embedded by 2KX .

37

We record here a related conjecture of Eisenbud, Koh, and Stillman as well as a partial answer proven by M.S. Ravi: Conjecture 5.2.3. [EKS] Let L be a very ample line bundle that embeds a smooth curve X. For each k there is a bound on the degree of L such that Seck X is ideal theoretically defined by the (k + 2) × (k + 2) minors of a matrix of linear forms. Theorem 5.2.4. [R] If deg L ≥ 4g + 2k + 3, then Seck X is set theoretically defined by the (k + 2) × (k + 2) minors of a matrix of linear forms. The following Proposition is proven by constructing forms as unions of hyperplanes; it amounts to some elementary combinatorics which we do not reproduce here: Proposition 5.2.5. Let X ⊂ Pn be a collection of reduced points in linear general kn position. If length X ≤ 2n − k+1 , k ≥ 0, then Seck X is set theoretically cut out by forms of degree k + 2. These statements provide enough evidence to make the following basic: Conjecture 5.2.6. Let L be an ample line bundle on a smooth variety X, k ≥ 1 fixed. Then for all n  0, Ln embeds X so that Seck X is ideal theoretically defined by forms of degree k + 2. Remark 5.2.7 An elementary calculation, e.g. via Macaulay [MAC] or see [W, 3.7], shows that if X is a curve with a 5-secant 3-plane, then any cubic which vanishes on SecX must vanish on that 3-plane. Hence SecX cannot be set theoretically defined by cubics. This should be compared to the fact that if X has a trisecant line, then X cannot be defined by quadrics. In particular, this shows that Green’s condition (N2 ) is not sufficient to guarantee that the secant variety is defined (even set theoretically) by cubics, e.g. if X is an elliptic curve embedded in P4 by a line bundle of degree 5. In fact in this case SecX is a quintic hypersurface, as the degree of the secant variety to a smooth curve of genus g and degree d in Pn is given by (d−1)(d−2) − g. 2 This implies that any uniform bound on the degree of a linear system that guarantees SecX is even set theoretically defined by cubics must be at least 2g + 4. 2 The same line of reasoning can be applied to higher secant varieties. Let Γ be a set of 2k + 3 general points in P2k+1 . If H 0 (P2k+1 , IΓk+1 (k + 2)) = 0, and if a curve has a 2k + 3-secant 2k + 1-plane, then Seck X cannot be set theoretically defined by forms of degree k + 2. The vanishing of H 0 above has been checked via Macaulay up to P9 . One can use earlier work to give a more geometric necessary condition for SecX to be defined as a scheme by cubics. Specifically, in Lemma 3.4.5 we saw that the ^ with the exceptional divisor E of the blow up of Pn along X is intersection of SecX isomorphic to Bl∆ (X × X). In the case that X is a smooth curve, this implies that if

38

^ → SecX be the blow up along X, then π −1 (p) ∼ we let π : SecX = X, p ∈ X. In fact, it is easy to verify (or see [B2, p.85]) that if X is embedded by a line bundle L, then π −1 (p) ∼ = X ⊂ PΓ(X, L(−2p)) where PΓ(X, L(−2p)) is identified with the fiber over p of the projectivized conormal bundle of X ⊆ Pn . Now, if SecX is defined as a scheme by cubics, then the ^ The restriction of this series to base scheme of OPfn (3H − 2E) is precisely SecX. PΓ(X, L(−2p)) is then a system of quadrics whose base scheme is X. In summary, if X is a smooth curve embedded by a line bundle L that satisfies (K2 ) and if SecX is scheme theoretically defined by cubics, then for every p ∈ X the line bundle L(−2p) is very ample and X ⊂ PΓ(X, L(−2p)) is scheme theoretically defined by quadrics. This implies that if one is to seek a uniform bound on deg L to imply that SecX is defined by cubics, deg L must be taken to be at least 2g + 4, the same bound encountered in Remark 5.2.7. 5.3. Construction of the Second Flip Suppose as above that X satisfies (K2 ), is smooth, and contains no lines and no plane quadrics. Suppose further that SecX is scheme theoretically defined by cubics C0 , . . . , Cs2 , and that SecX satisfies (K3 ). It is natural to expect condition (K3 ) to be satisfied by sufficiently large embeddings of X; notice that in Proposition 5.2.2, the cubics that at least set theoretically define the secant variety satisfy (K3 ) [JPW, 3.19]. Under these hypotheses, we construct a second flip as follows: We know that OM2 (3H − 2E) is globally generated by the discussion above; hence this induces a morphism ϕ e2 : M2 → Ps2 which agrees with the map given by the cubics ϕ2 : n s2 P 99K P on the locus where M2 and Pn are isomorphic. By Remark 3.2.8, ϕ e2 is a birational morphism. Therefore we can construct a flip simply by blowing up M2 along the exceptional locus of ϕ e2 and blowing back down as in Example 4.1.1. We wish first to identify the exceptional locus of ϕ e2 . It is clear that ϕ e2 will collapse the image of a 3-secant 2-plane to a point, hence the exceptional locus must contain the image of Sec2 X. However by Remark 3.2.8, we know that the rational map ϕ2 is an isomorphism off of the trisecant variety to the secant variety Sec13 (SecX). This motivates the following Lemma 5.3.1. Let X ⊂ Pn be an irreducible variety. Assume that Seck X is scheme theoretically defined by forms of degree ≤ 2k + 1. Then Seck X = Sec1k+1 (Seck−1 X) as schemes. Proof: First, choose a (k + 1)-secant k-plane M . M then intersects Seck−1 X in a hypersurface of degree k + 1, hence every line in M lies in Sec1k+1 (Seck−1 X). As Seck−1 X is reduced and irreducible, Seck X ⊆ Sec1k+1 (Seck−1 X) as schemes.

39

Conversely, choose a line L that intersects Seck−1 X in a scheme of length at least k + 1. It is easy to verify that Seck X is singular along Seck−1 X, hence every form that vanishes on Seck X must vanish 2k + 2 times on L. By hypothesis, however, Seck X is scheme theoretically defined by forms of degree ≤ 2k + 1, hence each of these forms must vanish on L. 2 Remark 5.3.2 Note that this implies that if Seck X satisfies (Kk+2 ) and if Seck+1 X is scheme theoretically defined by forms of degree ≤ 2k + 3, then the map ϕk+1 given by the forms defining Seck X is an isomorphism off of Seck+1 X. 2 Lemma 5.3.3. Let X ⊂ Pn be an irreducible variety whose embedding is (2k + 4)very ample. Assume that Seck X satisfies (Kk+2 ) and that Seck+1 X = Sec1k+2 (Seck X) as schemes. Let Γ be the closure of the graph of ϕk+1 with projection π : Γ → Pn . If a is a point in the closure of the image of ϕk+1 and Fa ⊂ Γ is the fiber over a then π(Fa ) is one of the following: (1) a reduced point in Pn \ Seck+1 X (2) a (k + 2)-secant (k + 1)-plane (3) contained in a linear subspace of Seck X Proof: The first and third statements follow directly from Lemma 5.3.1 and Proposition 3.2.2. For the second, note that a priori π(Fa ) could be any linear space intersecting Seck X in a hypersurface of degree k + 2. However, the (2k + 4)-very ample hypothesis implies that the intersection of two (k + 2)-secant (k + 1)-planes, if nonempty, must lie in Seck X. Hence if there is a linear subspace intersecting Seck X in a hypersurface of degree k + 2, it must be k + 1-dimensional. 2 These two lemmas should be viewed as the first steps towards constructing a sequence of flips analogous to those of Thaddeus. We return to the case k = 1: Theorem 5.3.4. Let X ⊂ Pn , n ≥ 5, be a smooth, irreducible variety that satisfies (K2 ). Assume that the following conditions are satisfied: (1) The embedding of X is 6-very ample (2) Sec1 X satisfies (K3 ) (3) Sec2 X = Sec13 (Sec1 X) as schemes (4) The projection of X into Pm , m = n − 1 − dim X, from any tangent space is such that the image satisfies (K2 ), contains no lines and contains no plane quadrics Then the morphism ϕ e2 : M2 → Ps2 induced by OM2 (3H − 2E) is an embedding off of 2 the transform of Sec X, and the fibers of ϕ e2 restricted to the transform of Sec2 X are isomorphic to P2 .

40

Remark 5.3.5 Note that if X is a smooth curve embedded by a line bundle of degree at least 2g + 5, then conditions 1 and 4 are automatically satisfied. If dim X ≥ 2, then the image of the projection from the space tangent to X at p is Blp (X) ⊂ Pm . Furthermore, by the discussion after Remark 5.2.7 the projections of X will be generated as a scheme by quadrics when Sec1 X is defined by cubics. 2 Proof:(of Theorem 5.3.4) s e2 . We need to identify f (ϕ e−1 Let a ∈ Ps2 be a point in the image of ϕ 2 (a)) ⊂ P −1 (see Diagram 5.1.2). We actually do this by studying the projection of (ϕ e2 ◦ h2 ) (a) s n f to P and then to P . fn is contained as a scheme in the total transBy Lemma 5.3.3, the projection to P form of the following: (1) a point in Pn \ Sec2 X (2) a 3-secant 2-plane to X (3) a linear subspace of Sec1 X. In the first case, there is nothing to show as the total transform of a point in Pn \ Sec2 X is simply a reduced point and the map ϕ e to Ps is an isomorphism in a neighborhood of this point. f2 → Pn be the For the other two cases, we need the following observation: Let M projection and let Fp be the fiber over p ∈ X. We have seen (Lemma 3.4.5) that Fp is the blow up of Pm along a copy of X (resp. Blp (X)). We denote this variety by X 0 ⊂ Pm , and the embedding of X 0 in Pm satisfies (K2 ) by hypothesis. The restriction of OM (3H − 2E1 − E2 ) to Fp can thus be identified with OBlX 0 (Pm ) (2H 0 − E 0 ), hence g 2 the only collapsing that occurs in Fp is that of secant lines to X 0 ⊂ Pm . Now, for some p ∈ X, suppose that a secant line L in Fp is collapsed by ϕ e2 ◦ h2 to a point. This secant line can be identified with a point of H2 X 0 , say [q, r] where q = r is allowed. It is now, however, not difficult to see that the proper transform of the 3-secant 2-plane to X ∈ Pn corresponding to the point [p, q, r] ∈ H3 X (Cf. Proposition 2.2.2) contains the line L ⊂ Fp above. Hence the only collapsing in the exceptional locus over a point p ∈ X is accounted for by the collapsing of 3-secant 2-planes. We now deal with the remaining cases. If the projection is a 3-secant 2-plane, then fn is a 3-secant 2-plane blown up at the three by our observation the projection to P points of intersection, and so the image in Ps is a P2 that has undergone a Cremona transformation. We have shown that the general fiber of ϕ e2 restricted to the transform of Sec2 X is two dimensional. Hence by upper semicontinuity every fiber is at least two dimensional. The only case left to deal with is 3), but in this case the fiber of ϕ e2 must lie in P(F ), which is impossible by the following Lemma 5.3.6. 2 Lemma 5.3.6. With hypotheses as in Theorem 5.3.4, the restriction of ϕ e2 to P(F ) has fibers of dimension at most 1.

41

Proof: Suppose that ϕ e2 has a fiber ϕ e−1 2 (a) ⊂ P(F ) of dimension d. s2 e−1 As ϕ e2 : M2 → P is the restriction of projection, this implies that f ◦ ϕ 2 (a) also 1 −1 has dimension d. As E2 → P(F ) is a P -bundle, (ϕ e2 ◦ h2 ) (a) ⊂ E2 has dimension d + 1, as does Y = π((ϕ e2 ◦ h2 )−1 (a)) ⊂ P(E ). As Y is the proper transform of a linear subspace of SecX ⊂ Pn , the dimension of ϕ(Y e ) can be d if and only if the general point of Y lies on a secant line contained fn → Pn , i.e. Y is in Y . Hence Y intersects the exceptional locus E of the blow up P the proper transform of a linear subspace of SecX of dimension d + 1 that intersects X; let x ∈ X be such a point. The intersection of Y with E therefore has dimension d. However, the system OM (3H − 2E1 − E2 ) collapses the transform of Y to a point, g 2 and so collapses a d dimensional subspace of the fiber over x to a point. However, by the remarks in the proof of Theorem 5.3.4, this is possible only if d ≤ 1. 2 As before, we show that the restriction of ϕ e2 to the transform of Sec2 X is a 2 X ⊂ M for the image ^ projective bundle. By a slight abuse of notation, write Sec 2 2 n of the proper transform of Sec X ⊂ P . Following our previous line of argument, we 2 X → H3 X. Note first the following: ^ define a map Sec 3 ∼ 2 Lemma 5.3.7. Let Z = ϕ e−1 2 (x) = P , x ∈ H X. Then OZ (H) = OP2 (2) and OZ (E) = OP2 (3).

Proof: As Z is a P2 that has undergone a Cremona transformation, E ∩ Z is a degenerate cubic in Z, hence OZ (E) = OP2 (3). Furthermore, the Cremona transformation P2 99K P2 is given in coordinates as [x, y, z] 7→ [xy, xz, yz], hence OZ (H) = OP2 (2). 2 2 X → G(2, n) whose image is H3 X. ^ Lemma 5.3.8. There exists a morphism Sec

Proof: 2 X be a point. Then p determines a unique 2-plane S in Sec 2 X by the ^ ^ Let p ∈ Sec p 0 above Proposition. For every p, we have then a homomorphism: H (M2 , OM2 (H)) → H 0 (M2 , OSp (H)) of rank 3, hence a point in G(2, n). The image of this morphism clearly coincides with the embedding of H3 X into G(2, n) described in Proposition 2.2.2. 2 As in Lemma 3.4.3, one can construct a morphism H3 X → Ps2 so that the com2 X → Ps2 . This is done by associating to every ^ position commutes with ϕ e2 : Sec 3 Z ∈ H X the rank 1 homomorphism: H 0 (M2 , OM2 (3H − 2E)) → H 0 (M2 , OM2 (3H) ⊗ OSZ ) where SZ is the plane in M2 associated to Z.

42

2 X with a P2 ^ Exactly as in Theorem 3.4.6, this allows the identification of Sec bundle over H3 X. Specifically, E2 = (ϕ e2 )∗ (OSec (2H − E)) is a rank 3 vector bundle ^ 2X 3 on H X and:

Proposition 5.3.9. With notation as above, 2X ∼ ^ Sec = PH3 X (E2 )

Hence blowing up Sec2 X along X and then along SecX resolves the singularities of Sec2 X. 2 f3 = BlP(E ) (M2 ). There is now a diagram: Let M 2 ~~ ~~ ~ ~~ ~

f2 M A

AA h2 AA AA A

fn P BB 

Pn

BB ϕe BB BB !

Ps

{{ {{ { { }{{ f

}} }} } } }~ }

f3 M

M2 C

CC ϕe CC 2 CC C!

Ps 2

M3 }

where, in order to complete the second flip, it is necessary to construct the morphisms indicated by dotted lines. As before, we construct the exceptional locus PH3 X (F2 ) = f3 = ZH + ZE1 + ZE3 . P(F2 ) ⊂ M3 . We write Pic M Lemma 5.3.10. Let p3 : E3 → H3 X be the composition E3 → P(E2 ) → H3 X. Then F2 = (p3 )∗ OE3 (4H − 3E1 − E3 ) is locally free of rank 2 X, M ) ^ n − 3dim X − 2 = codim (Sec 2

Proof: 2 X, M ). Further^ Each fiber Fx of p3 is isomorphic to P2 × Pt , t + 1 = codim (Sec 2 more H 0 (Fx , OFx (4H − 3E1 − E3 )) = H 0 (Pt , OPt (1)) follows easily from the fact that on M2 , if S ∼ e2 , then OS (H) = OP2 (2) = P2 is a fiber of ϕ and OS (E) = OP2 (3). 2 There is then a map E3 → P(F2 ) given by the surjection p∗3 F2 → OE3 (4H − 3E1 − E3 ) → 0 hence a diagram of exceptional loci:

43

w ww ww w w {w w

P(E )

HH HH HH HH H#

E2 H HH HH HH HH $ P(F )

v vv vv v v v{ v

P(E2 )

HH HH HH HH H$

v vv vv v v zv v

E3 I II II II II $ P(F2 ) u uu uu u u uz u

H2 X H3 X It is important to note that P(F2 ) ∼ = P(p3 )∗ OE3 (4H − 3E1 − E3 + k(3H − 2E1 )) for all k ≥ 0 as the direct image on the right will differ from F2 by a line bundle. Hence for all k ≥ 0 there is a morphism E3 → P(F2 ) induced by the surjection p∗3 (p3 )∗ OE3 (4H − 3E1 − E3 + k(3H − 2E1 )) → OE3 (4H − 3E1 − E3 + k(3H − 2E1 )) One can now repeat almost verbatim (4.4.3)-(4.4.6) to construct the second flip. f3 → M3 is given by the linear system associated to The map M OM ((3k − 2)H − (2k − 1)E1 − E3 ) = OM ((4H − 3E1 − E3 ) + (k − 2)(3H − 2E1 )) g g 3 3 for k  0. The only difficulty is the numerics of the vanishing shown in the proof of Proposition 4.4.6. For this, one defines Lk = O((3k − 2)H − (2k − 1)E1 − E3 ) −1 n−3r−1 and writes (in the notation of the proof) B ⊗ KM plus a big and nef Q g as Lα divisor, where α =

3

2k+n−r−1 . 2n−6r−2

Lemma 5.3.11. As defined above M3 is smooth. Proof: M3 is normal by Lemma 3.2.5. Let Z ∼ = P2 be a fiber of h3 over a point p ∈ P(F2 ). Z is a fiber of a P2 ×Pt bundle over H3 X, hence the normal bundle sequence becomes: M 0→ OP2 → NZ/M → OP2 (−1) → 0 g 3 It is easy to see that this sequence splits, and an elementary calculation similar to that in Proposition 4.5.1 gives H 1 (Z, S r NZ/M ) = 0 and g 3 H 0 (Z, S r NZ/M ) = S r H 0 (Z, NZ/M ) g g 3 3 for all r ≥ 1. By a natural extension of Proposition 4.5.1 (given in [AW, 2.4] with the same proof) M3 is smooth. 2 5.4. Construction of Further Flips In this section we outline, without proofs, how further flips may be constructed. To ease notation, say X ⊂ Pn satisfies condition (K2` ) if Seci X satisfies condition (K2+i ) for 0 ≤ i ≤ `; hence (K20 ) = (K2 ). Inductively, assume that the first k flips have been constructed as above, i.e. through the space Mk+1 . If the hypotheses of Lemma 5.3.3 are satisfied, there is a

44

morphism ϕ ek+1 : Mk+1 → Psk+1 induced by OMk+1 ((k + 2)H − (k + 1)E) which is an isomorphism off of the transform of Seck+1 X. In order to control ϕ ek+1 on the transform of Seck+1 X, an observation like that in the proof of Theorem 5.3.4 is needed. Denote recursively B1 = BlX (Pn ), Bi = fn , B2 = M f2 . Let πk : Bk → Pn be the composition of (Bi−1 ); hence B1 = P BlSec ^ i−1 X blow ups. Consider the diagram in the case k = 2: B3 B || || | | |~ |

~~ ~~ ~ ~~ ~

fn P BB π1



f2 M A

BB ϕe BB BB !

AA AA AA A

{{ {{ { { {} {

BB BB BB B

}} }} } } }~ }

M2 C

f3 M A

CC ϕe CC 2 CC C!

AA AA AA A

{{ {{ { { {} {

M3 C

CC ϕe CC 3 CC C!

Pn Ps Ps 2 Ps 3 The idea is now to lift ϕ e3 back to B3 and apply the linear system |4H − 3E1 − 2E2 − E3 | For simplicity, assume that X ,→ Pn is given by a complete linear system associated to a line bundle L. One can derive the following facts: (1) Let x ∈ X ⊂ Pn . (a) π1−1 (x) ∼ = PH 0 (X, L(−2Ex )) ∼ = Pn−dim X−1 , where Ex is the exceptional divisor of X 0 = Blx (X) (b) π1−1 (x) ∩ P(E ) ∼ = X 0 , hence π −1 (x) ∼ = BlX 0 (Pn−dim X−1 ) 2

2X ∼ ^ (c) π1−1 (x) ∩ Sec = Sec1 X 0 ⊆ PH 0 (X, L(−2Ex )), hence n−dim X−1 π3−1 (x) ∼ )) = BlSecX ^0 (BlX 0 (P

(2) Let x ∈ Sec1 X \ X (a) π2−1 (x) ∼ = PH 0 (X, L(−2(Ep + Eq ))) ∼ = Pn−2dim the secant line containing x. 2X ∼ ^ (b) π2−1 (x) ∩ Sec = X 00 = Blp∪q (X), hence π −1 (x) ∼ = BlX 00 (Pn−2dim X−2 )

X−2

where p, q ∈ X span

3

(3) Let x ∈ Sec X \ Sec1 X. Then π3−1 (x) ∼ = PH 0 (X, L(−2(Ep + Eq + Er ))) ∼ = n−3dim X−3 P where p, r, q ∈ X span the 2-plane containing x. In the case X is a curve, this is a statement of “Terracini Recursiveness” given in [B, 1.4] and [B3, 3.6b]. To see the relevance of these facts, restrict for the moment to the case X is a curve, hence X ∼ = X0 ∼ = X 00 . Then the restriction of the system OB3 (4H − 3E1 − 2E2 − E3 ) to: 2

45

n−2 (1) π3−1 (x), x ∈ X is the system O(3H−2E1 −E2 ) on the space BlSecX )) ^ (BlX (P where H is the hyperplane class from Pn−2 and E1 , E2 are the exceptional divisors from the blow ups (2) π3−1 (x), x ∈ SecX \ X is the system O(2H − E) on the space BlX (Pn−4 ) To control OB3 (4H − 3E1 − 2E2 − E3 ) on the exceptional loci, we restrict to various fibers as in the proof of Theorem 5.3.4. Under sufficient hypotheses (as in Theorem 5.3.4), these restrictions are previously studied systems. Hence one can continue constructing flips recursively. There an important fact that should be noted: In general Hk X is singular if k ≥ 4 and dim X ≥ 3. This will not cause a problem for the map ϕ e3 , however when we get a P3 -bundle over a singular space, H4 X, blowing up M3 along this space need not yield a smooth variety. Hence it is only clear that this construction will proceed nicely in the case dim X = 1, 2. An immediate corollary of the above facts is

Corollary 5.4.1. Let X ⊂ PΓ(X, L) be a smooth curve satisfying (K2i−1 ) and assume that Seci X is defined set (resp. scheme) theoretically by forms of degree i + 2. Then (1) L(−2Z) is very ample for all Z ∈ Hi X (2) If Xj ∼ = X denotes the image of the embedding X ,→ PΓ(X, L(−2Zj )), Zj ∈ j H X, then Seck Xj is set (resp. scheme) theoretically defined by forms of degree k + 2 for k ∈ {0, . . . , i − j}, j ≤ i. We conclude with the general flip construction for curves. It should be noted that the hypotheses are at present not well understood. Theorem 5.4.2. Let X ⊂ Pn be a smooth, irreducible curve embedded by a line bundle L. Assume the following: (1) X satisfies (K2i ) (2) deg L ≥ 2g + 3 + 2i (3) Seck X = Sec1k+1 (Seck−1 X) for k ≤ i + 1 −1 (4) If x ∈ Seck X \ Seck−1 X then the associated copy of X ⊂ πk+1 (x) satisfies (K2i−k−1 ) Then i + 1 flips exist, i.e. the space Mi+2 can be constructed, with the following properties: (1) The maps ϕ ek : Mk → Psk are induced by OMk ((k + 1)H − kE) for k ≤ i + 1 fk → Mk are induced by Og ((k + 1)H − kE1 − Ek ) for (2) The maps hk : M Mk ] k ≤ i + 1 and the map hi+1 : Mi+2 → Mi+2 is induced by OM ^ (((i + 2)r − (i + 1))H − ((i + 1)r − i)E1 − Ei+2 ) i+2

for r  0 (3) The spaces Mk are smooth (4) The exceptional loci of the k th flip diagram are projective bundles over Hk+1 X: PEk = Pϕ ek∗ OSec (kH − (k − 1)E) ^ kX

46

and PFk = Ppk+1∗ OEk+1 ((k + 2)H − (k + 1)E1 − Ek+1 ) of dimension k and n − k − 1 respectively, where pk is the map Ek → Hk X (5) The ample cone of Mk is bounded by OMk (kH − (k − 1)E) and OMk ((k + 1)H − kE) for k ≤ i + 1 ~~ ~~ ~ ~ ~ ~

f2 M A

AA h AA 2 AA A

fn P BB 

Pn

BB ϕe BB BB !

Ps

{ {{ {{ { }{ {

}} }} } } } ~}

] M i+2

f3 M @

@@ h @@ 3 @@ @@

M2 C

CC ϕe CC 2 CC C!

Ps 2

···

|| || | || |} |

{{ {{ { {{ {} { · · ·E EE EEϕei+1 EE EE "

Mi+2

v vv vv v v v{ v

Psi+1

47

FF FFhi+2 FF FF "

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