THE HILBERT SCHEME OF A PLANE CURVE SINGULARITY AND

THE HILBERT SCHEME OF A PLANE CURVE SINGULARITY AND THE HOMFLY POLYNOMIAL OF ITS LINK

arXiv:1003.1568v1 [math.AG] 8 Mar 2010

ALEXEI OBLOMKOV AND VIVEK SHENDE

A BSTRACT. The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We conjecture that the generating function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreducible singularities y k = xn , whose links are the k, n torus knots, and for the singularity y 4 = x7 − x6 + 4x5 y + 2x3 y 2 , whose link is the 2,13 cable of the trefoil.

1. I NTRODUCTION Let X be a quasi-projective algebraic scheme, and let X [⋆ ] be the set of its zero dimensional subschemes of finite length. A theorem of Grothendieck [G] implies that X [⋆ ] carries a natural topology and admits a scheme structure; thus equipped, it is called the Hilbert scheme of points. Although X [⋆ ] naturally decomposes into finite dimensional pieces X [l] parameterizing subschemes of length l, we prefer to view X [⋆ ] as a single space carrying a function l : X [⋆ ] → N giving at a point [z ] ∈ X [⋆ ] the length of the subscheme z ⊂ X. For a smooth curve C, the Hilbert scheme C [⋆ ] parameterizes unordered tuples of not-necessarilydistinct points in C. It can be seen to be smooth from the fundamental theorem of symmetric functions. Its graded Euler characteristic is given by the formula: Z ∞ ∞ X X 2l [l] 2l q 2l χ(Syml (C)) = (1 − q 2 )−χ(C) q χ(C ) = (1) q dχ = C [⋆ ]

l=0

l=0

The integral with respect to Euler characteristic is defined by cutting up the space into strata on which the integrand is constant, and summing the product of the integrand with the Euler number of the strata. Thus the first equality holds by definition, the second is true because C [l] = Syml (C), and the third is valid for any sufficiently nice topological space C. There is always a map X [n] → Symn (X), but it is not an isomorphism in general. For a smooth surface S, Symn (S) is mildly singular, S [n] is smooth, and the map S [n] → Symn (S) is a crepant resolution of singularities. G¨ottsche [Gt] has computed the graded Euler characteristic of S [⋆ ] : Z ∞ Y 2l (2) q dχ = (1 − q 2k )−χ(S) S [⋆ ]

k=1

Vafa and Witten [VW] note that a certain physical duality induces an isomorphism between the Fock space on the cohomology of S and the cohomology of S [⋆ ] , thus explaining the formula. This suggested that there should be a geometric construction of a Heisenberg algebra structure on H∗ (S [⋆ ] ); this was provided by Grojnowski [Gr] and Nakajima [N]. 1

Consider now a possibly singular curve C in a smooth surface X. Equation (1) allows us to reduce the computation of the graded Euler characteristic of C [⋆ ] to a neighborhood of the singular locus of C. Let Csm be the smooth locus of C, and Cp1 , . . . , Cpr the formal germs of C at the various singularities. Stratify C [⋆ ] by the length supported at the singular points to deduce: Z Z Y Z Y Z 2l 2l 2l 2 −χ(Csm ) q dχ = q dχ × q dχ = (1 − q ) q 2l dχ C [⋆ ]

Csm [⋆ ]

i

i

Cpi [⋆ ]

Cpi [⋆ ]

The topology of the pair (C, X) near a singularity of C is studied via knot theory [AGV, EN, M, Z2]. Consider the intersection of C and a small 3-sphere around a singular point in C. This is a collection of oriented circles in S 3 , one for each analytic local branch of the singularity; it is called the link of the singularity. For example, the curve y k = xn intersects a sphere around the origin in the k, n torus link. Milnor [M] has studied links of hypersurface singularities, and shows in particular that the link of any singularity is fibred. Eisenbud and Neumann [EN] describe how to pass from the combinatorics of the Puiseux data of the singularity to the topology of the link. Denote by P(L) the HOMFLY polynomial of an oriented link L ⊂ S 3 . It is an element of Z[a±1 , (q − q −1 )±1 ], and may be computed from the relations a P(") − a−1 P(!) = (q − q −1 ) P(H) P( ) = 1

(3) (4)

The meaning of Equation (3) is that three link diagrams which are identical away from a small neighborhood, and are as depicted within it, have HOMFLY polynomials satisfying the given skein relation. Equation (4) normalizes the HOMFLY polynomial by requiring its value on the unknot to be 1. It is not obvious that these relations define a function on diagrams, let alone knots, but it is true [HOMFLY]. The topological meaning of this invariant is unknown. Conjecture 1. Let C be a curve in a smooth surface, C itself smooth away from points pi . Let Li be the link of C at pi , and let µi be the Milnor number of the singularity at pi . Z Y q 2l dχ = (1 − q 2 )−χ(C) [(q/a)µi P(Li )]a=0 i

C [⋆ ]

This statement has a certain physical significance which we now sketch. Recall from Kawai and Yoshioka [KY] that if L is a primitive very ample bundle on a K3 surface such that L2 = 2g − 2, Z Z ∞ Y 1 −1 2 2l (5) (q − q ) q dχ = Coeff pg n 20 2 (1 − p ) (1 − q pn )2 (1 − q −2 pn )2 n=1 C∈|L| C [⋆ ]

Pandharipande and Thomas [PT3], in their study of curve counting in surfaces and three-folds, use Serre duality to establish the genus expansion for reduced C: Z g X 2l 2g−2 nh (C)(q −1 − q)2h−2 (6) q dχ = q C [⋆ ]

h=g

In the formula, g, g are the arithmetic and geometric genera of C, and the nh (C) are integers. Upon combining these formula to obtain Z X g ∞ Y 1 −1 2h 2g−2 nh (C)(q − q) dχ = Coeff pg (7) q n 20 2 (1 − p ) (1 − q pn )2 (1 − q −2 pn )2 n=1 h=g C∈|L|

2

Pandharipande and Thomas observe that Equation 7 matches the prediction of Katz, Klemm, and Vafa [KKV] if the nh (C) are the conjectural Gopakumar-Vafa numbers measuring the number of genus h curves contributed by C [GV]. Thus Conjecture 1 suggests a relation between the string theory used to define BPS numbers of a locally planar curve, and Chern-Simons theory near its singularities [W]. We turn to the question of how the HOMFLY polynomial, rather than merely its a → 0 limit, may be recovered from the Hilbert scheme. It is necessary to introduce another grading. Let us fix [⋆ ] attention on the formal germ Cp of the curve at a point; we will define a function m : Cp → N. [⋆ ] ˆ C,p be the Let [Z] ∈ Cp be the point corresponding to the subscheme Z ⊂ Cp , and let IZ ⊂ O ideal of Z in the complete local ring. Then m([Z]) is the minimal number of generators of IZ . m : Cp[⋆ ] → N [Z] 7→ dim IZ ⊗OˆC,p C Conjecture 2. Let p ∈ C be a point on a locally planar curve. Let Cp be the formal germ of C at p, let LC,p be the link of C at p, and let µ be the Milnor number of the singularity. Z µ 2 P(LC,p ) = (a/q) (1 − q ) q 2l (1 − a2 )m−1 dχ Cp [⋆ ]

Conjecture 2 implies Conjecture 1 by a stratification argument. The remainder of the article presents evidence for the conjectures, which we briefly itemize below. • Setting a = −1 in Conjecture 2 leaves the formula Z −µ 2 ∇(LC,p ) = q (1 − q ) q 2l dχ ∗ OC,p /OC,p

∗ where ∇ is the Alexander polynomial in a suitable normalization, and OC,p /OC,p parameterizes functions up to multiplication by invertible functions, or in other words, ideals with one generator. After some technical rearrangements, this formula follows from a theorem of Campillo, Delgado, and Gusein-Zade. Details appear in Section 3. • The skein relation exhibits the invariance the HOMFLY polynomial under the transformation q → −q −1 ; in fact, P ∈ Z[(q − q −1 )±1 , a±1 ]. It is not evident from their expressions that our integrals enjoy the same property; nonetheless this was verified by Pandharipande and Thomas for the integral in Conjecture 1, and we explain in Section 4 how to extend their methods to the integral in Conjecture 2. • Singularities of the form y k = xn carry a torus action which lifts to the Hilbert scheme. In the case gcd(k, n) = 1, the fixed points are isolated, and we count them in Section 5 to calculate the integral in Conjecture 2. Jones has calculated the HOMFLY polynomial of the corresponding (k, n)-torus knot, and the formulae match. • Section 6 describes, in the case of unibranch singularities, a stratification of the Hilbert scheme of points via the semigroup of the singularity. This is closely related to Piontkowski’s work on computing the cohomology of compactified Jacobians [Pi]. In Section 7 we compute explicitly the strata of the Hilbert scheme of the singularity with complete local ring C[[t4 , t6 + t7 ]], and verify that the generating function of its Euler characteristics matches the HOMFLY polynomial of the (2,13) cable of the right-handed trefoil knot.

3

Remark. In Equations (1) and (2), it is natural to promote the Euler characteristic to the virtual Poincar´e polynomial. The same is true on the left hand side of Conjecture 2; the integral now computes what may be regarded as the homology of a bigraded space. On the other hand, the HOMFLY polynomial is known to arise as the Euler characteristic of the cohomology of a bigraded complex [KR]. We will state a homological version of Conjecture 2 in a subsequent article [ORS]. Acknowledgements. We are grateful to Rahul Pandharipande for suggesting this area of study – during a class for which the notes may be found on his website [P] – and for advice throughout the project. We have also enjoyed discussions with Margaret Doig, Eduardo Esteves, Paul Hacking, Jesse Kass, Jacob Rasmussen, Giulia Sacc`a, Sucharit Sarkar, Richard Thomas, and Kevin Wilson. A.O. was partially supported by NSF grant DMS-0111298. 2. S MOOTH

POINTS , NODES , AND CUSPS

We illustrate the conjecture with some elementary examples. Denote by C2,n the formal germ at the origin of the curve cut out by y 2 = xn , and by O2,n its ring of functions. The link of this singularity is the 2,n torus link T2,n . The first few of these: T2,0 =

T2,1 =

T2,2 = )

T2,3 = &

Computing P(T2,n ) is an elementary exercise in the skein relation: smoothing a crossing yields T2,n−1 and switching a crossing gives T2,n−2 . This yields the recurrence P(T2,n ) = −a(q − q −1 )P(T2,n−1) + a2 P(T2,n−2 ) T2,1 is the unknot, and T2,0 is two unlinked circles. It is immediate from the skein relation that the HOMFLY polynomial of n unlinked circles is ((a − a−1 )/(q − q −1 ))n−1 . Thus:

(9)

a − a−1 q − q −1 P(T2,1 ) = 1

(10)

P(T2,2 ) = −a(q − q −1 ) +

(8)

(11)

P(T2,0 ) =

a3 − a q − q −1 P(T2,3 ) = a2 q 2 + a2 q −2 − a4

We now compute the integral of Conjecture 2 for n = 1, 2, 3. Example 3. As y 2 = x is smooth at the origin, the Milnor number is µ = 0. The ring O2,1 = C[[t]] has ideals (ti ) for i ∈ N. Then the conjecture asserts 1 = (a/q)0 (1 − q 2 )

∞ X

q 2i

i=0

Example 4. At the origin, y 2 = x2 has a node, so the Milnor number is µ = 1. The ideals of the ring O2,2 = C[[t1 , t2 ]]/(t1 t2 ) are: (1) ∗ (tx1 + λti−x 2 ) where i ≥ 2, x = 1 . . . i − 1, and λ ∈ k x i−x+1 (t , t ) where i ≥ 1, x = 1 . . . i 4

In each case the variable i gives the colength of the ideal. Each component of the space of ideals of the second type is C∗ and thus has Euler characteristic zero. Thus the conjecture asserts ! ∞ 3 X a − a −a(q − q −1 ) + = (a/q)1 (1 − q 2 ) 1 + (1 − a2 ) iq 2i q − q −1 i=1 Example 5. At the origin, y 2 = x3 has a cusp, so the Milnor number is µ = 2. The ideals of the ring O2,3 = C[[t2 , t3 ]] are: (1) (ti + λti+1 ) for i ≥ 2 and all λ ∈ C (ti+1 , ti+2 ) for i ≥ 1 Each component of the space of ideals of the second type is C and thus has Euler characteristic 1. Thus the conjecture asserts ! ∞ ∞ X X q 2i a2 q 2 + a2 q −2 − a4 = (a/q)2 (1 − q 2 ) 1 + q 2i + (1 − a2 ) i=2

3. S PECIALIZATION

TO THE

A LEXANDER

i=1

POLYNOMIAL

In this Section, we show Conjecture 2 holds in the limit a = −1: Proposition 6. Let C be the germ of a plane curve singularity, and let LC its link. Then Z µ 2 P(LC )|a=−1 = lim (a/q) (1 − q ) q 2l (1 − a2 )m−1 dχ a→−1

C [⋆ ]

Both sides of the above equality have simpler expressions. The left hand side is the AlexanderConway polynomial, denoted ∇(LC ), as can be seen by specializing the skein relations:1 ∇(!) − ∇(") = (q − q −1 )∇(H) ∇( ) = 1 Since the integrand on the right hand side vanishes unless m = 1, Proposition 6 is equivalent to: Z 2 −µ q 2l dχ (12) ∇(LC ) = (1 − q )(−q) O/O ∗

where at a point of O/O∗ represented by a function f , the function l takes the value dim O/f O. We will derive Equation (12) from a theorem of Campillo, Delgado, and Gusein-Zade which recovers the Alexander polynomial from the ring of functions [CDG]. Theorem 7. (Campillo–Delgado–Gusein-Zade [CDG]) Let C be germ of a plane curve singuLthe b larity; let L be the associated link. Fix a normalization OC ֒→ i=1 C[[zi ]]. Define ν = (ν 1 , . . . , ν b ) : O → (N × C∗ )b

1

It is more common to write q = t1/2 or z = q − q −1 . 5

in which ν i records the degree and coefficient of the leading term in zi . Let νi be the analogous functions recording only the degree of the leading term. Then if b > 1, the multivariate Alexander polynomial is given by Z (13) ∆L (t1 , . . . , tb ) = tν11 · · · tνb b dχ ν(O)/C∗

For b = 1, the above formula gives ∆L (t1 )/(1 − t1 ). The Alexander polynomials have been normalized so ∆L ∈ 1 + (t1 , . . . , tb )Z[t1 , . . . , tb ]. The multivariate Alexander polynomials occuring in the theorem are defined in terms of the universal abelian cover of the link complement, whereas the usual univariate Alexander polynomial may be defined in terms of the infinite cyclic cover. The univariate Alexander polynomial may be recovered by evaluating at t = t1 = · · · = tn and multiplying by normalization factors. For multibranch hypersurface singularities, Milnor shows that ∆L (t, . . . , t) = (1 − t)∆L (t), where the Alexander polynomial on the right is the one-variable Alexander polynomial [M, Lemma 10.1]. Therefore, Z P (14) ∆L (t) = (1 − t) t νi dχ ν(O)/C∗

where ∆L (t) is the univariate Alexander polynomial, normalized so ∆L (t) ∈ 1 + tZ[t]. Lemma 8. Let C be the germ of a plane curve singularity; let L be its link and µ its Milnor number. Let ∆L (t) be the Alexander polynomial, normalized by ∆L (t) ∈ 1 + tZ[t]. Let ∇L (q) be the Alexander-Conway polynomial normalized by the skein relation given above. ∆L (q 2 ) = (−q)−µ ∇L (q) Proof. It is well known that ∆L (q 2 ) = ±q −n ∇L (q). Since ∇L (q) = ∇L (−1/q), the integer n must be the degree of the Alexander polynomial, which Milnor shows to be µ [M]. To resolve the sign ambiguity, recall that the link of a plane curve singularity may be realized as the closure of a braid that such “positive links has in which only positive crossings (!) appear. Van Buskirk Phas shown −1 i positive Conway polynomials [vB],” meaning ∇L (q) = ni (q − q ) for ni ∈ N. 2  We rewrite Equation (14) as

(15)

2

∇L (q) = (1 − q )(−q)

−µ

Z

q2

P

νi

dχ

ν(O)/C∗

Proof of Proposition 6. It remains to match the integrals in Equations (14) and (15). Restrict ν : O/O∗ → ν(O)/C∗ . We will show in Lemma 9 that the fibres of this map are affine spaces3 – ∗ so we may pull the integral with respect to Euler Pcharacteristic back to the space O/O – and in ∗ Lemma 10 that as functions on O/O , we have νi (f ) = dim O/f O.  2

In van Buskirk’s paper, much more precise conditions on the ni are given. We expect that in fact all these affine spaces have the same dimension. This, however, is not necessary for our application: it suffices to cut up the base ν(O)/C∗ according as the dimension of the fibre, compute Euler characteristic, and then sum. 3

6

Lemma 9. The fibres of ν : O/O∗ → ν(O)/C∗ are affine spaces. Proof. Let O1∗ ⊂ O be the invertible functions with leading term 1. Then O∗ = C∗ × O1∗ , and the fibres of the map in statement agree with the fibres of ν : O/O1∗ → ν(O). Any [f ] ∈ O/O1∗ has a representative in O of the form X f˜ = z ν(f ) + λi z i i>ν(f ) i∈ν((f / ))

Here, i, ν(f ) are multi-indices, and i > ν(f ) means that all components of ν(f ) − i are nonnegative, and at least one is strictly positive. Such a representative is unique: given two such, f˜ and f˜′ , we have ν(f˜ − f˜′ ) ∈ / ν((f )) unless f˜ = f˜′ . Let S ⊂ O be the set of these representatives. Fix some v0 ∈ ν(O) ⊂ (N × C∗ )b , and observe L that Sv0 = S∩ν −1 (v0 ) lies inside the finite dimensional affine space of all elements of bi=1 C[[zi ]] with the appropriate leading term and no other terms with degrees outside Nb \ ν((f )). As the line through any two elements of Sv0 remains inside Sv0 , we see Sv0 is an affine space.  Lemma 10. Let A ֒→ B be rings and let f ∈ A be a non zero divisor in both A and B. If B/A, A/f A, and B/f B have finite length as A-modules, then A/f A and B/f B have equal length. Proof. Consider the diagram 0 −−−→ f A −−−→   y

A −−−→ A/f A −−−→ 0     y y

0 −−−→ f B −−−→ B −−−→ B/f B −−−→ 0 The Snake Lemma provides the long exact sequence 0 → ker(f A → f B) → ker(A → B) → ker(A/f A → B/f B) → f B/f A → B/A → (B/f B)/(A/f A) → 0 Note that the first two modules in each line are abstractly isomorphic (indeed, the ones on the first line are zero). Since all these modules have finite length, the alternating sum of the lengths is zero, and hence ker(A/f A → B/f B) and (B/f B)/(A/f A) have the same length. But then A/f A and B/f B have the same length.  For computations, the following is useful. Corollary 11. Let Γ ⊂ N be a semigroup with #N \ Γ < ∞. For i ∈ Γ, #Γ \ (i + Γ) = i. More generally, let Γ ⊂ ∆ ⊂ N, and suppose i + ∆ ⊂ Γ. Then #Γ \ (i + ∆) = i − #∆ \ Γ Proof. Consider C[Γ], the ring with generators {xγ }γ∈Γ and relations xγ1 xγ2 = xγ1 +γ2 . Now apply Lemma 10 to C[Γ] ֒→ C[x] and xi ∈ C[Γ] to see #Γ \ (i + Γ) = i. The final statement follows from #Γ \ (i + ∆) + #(i + ∆) \ (i + Γ) = #Γ \ (i + Γ) = i and #(i + ∆) \ (i + Γ) = #∆ \ Γ.  4. T HE

GENUS EXPANSION

The HOMFLY polynomial lies in Z[a±1 , (q − q −1 )±1 ]; in particular, the skein relation defining it is manifestly invariant under the involution q → −1/q. We will show in this Section that the same properties hold for the quantity on the right hand side of Conjecture 2. In the a → 0 limit of Conjecture 1, this has already been done by Pandharipande and Thomas [PT3, Appendix B]. 7

Their approach ultimately rests on Serre duality and the Abel-Jacobi map, and works without modification for the series in Conjecture 2 once note is taken of the following fact from commutative algebra [E]: Lemma 12. Let O = C[[x, y]]/(f ) be reduced, let M be a torsion free O module, and let M D = Hom(M, O). Then M and M D have the same number of generators. Proof. Write A = C[[x, y]] and consider M as an A-module. M is torsion free, so any element of A which does not reduce to zero in O is a nonzero divisor on M. Thus depth(x,y) M ≥ 1 and by the Auslander-Buchsbaum theorem, proj. dim(M) = depth(x,y) C[[x, y]] − depth(x,y) M ≤ 2 − 1 = 1 On the other hand, any resolution must have at least two terms, since M is not a free A-module. Fix a minimal two-term resolution of M. Since M is rank zero as an A-module the two terms have the same rank; by minimality, this is the number of generators of M. Then the sequence 0 → Am → Am → M → 0 gives rise to 0 = HomA (M, A) → Am → Am → Ext1A (M, A) → 0 Since Ext1A (M, A) = HomO (M, O) = M D , we see that M D has at most as many generators as M. O is by hypothesis a hypersurface and hence Gorenstein. So (M D )D = M, and we may repeat the argument to see that M D has at least as many generators as M.  We now apply the methods of Pandharipande and Thomas [PT3]. The first step is to trade ideals in the complete local ring of the singularity for sheaves on a complete curve. Any locally planar singularity may occur on a rational curve C smooth away from a single point p; we will write Cp for the germ of this curve at p. The arithmetic genus of C is the delta invariant δ of Cp . As always we write l for the function measuring length on C [⋆ ] . We define m : C [⋆ ] → N at a subscheme Z ⊂ C to be the minimal number of generators of its ideal sheaf restricted to Cp .4 Lemma 13. Z

2l

2 m−1

q (1 − a )

2 b−2

dχ = (1 − q )

Z

q 2l (1 − a2 )m−1 dχ

[⋆ ]

C [⋆ ]

Cp [⋆ ]

Proof. Consider the map (C \ p)[⋆ ] × Cp → C [⋆ ] which takes the (disjoint) union of a scheme supported away from p and a scheme supported at p. The map is a bijection with constructible inverse. Denote by lp , lp , lC the length function on the three spaces. For X supported at p, Y supported away from p, and Z their union, we have lp ([X]) + lp ([Y ]) = lC ([Z]). By definition, m([Z]) = m([X]). Thus we compute Z Z Z 2lC 2 m−1 2lp q (1 − a ) dχ = q dχ × q 2lp (1 − a2 )m−1 dχ C [⋆ ]

C\p[⋆ ]

[⋆ ]

Cp

By Equation (1), the first term in the latter product is (1 − q 2 )−χ(C\p) .



4Presumably it is equivalent to let m give the rank of the first term in a free resolution of the ideal sheaf as a module over the structure sheaf of a smooth surface containing C.

8

Let I be an ideal sheaf on C. Then the sequence 0 → I → OC → OZ → 0 passes by Hom C (·, OC ) into 0 → OC → Hom C (I, OC ) → Ext 1C (OZ , OC ) → 0 This induces an isomorphism of C [⋆ ] with the moduli space of pairs [PT3, Appendix B] Pairs(C) = {OC → M | M a rank 1 torsion free sheaf} The isomorphism matches length l schemes to sheaves of holomorphic Euler characteristic l+1−δ. Forgetting the section sends the pairs space to the compactified Picard scheme Pic(C) = {rank 1 torsion free sheaves} The dimension of this space is the arithmetic genus δ of C. The fibre over a sheaf M is PH0 (C, M). Let h0 , e be the functions on Pic(C) giving the dimension of the space of global sections and the holomorphic Euler characteristic. Z Z 2l 2 m−1 2δ−2 q 2e (1 − a2 )m−1 h0 dχ q (1 − a ) dχ = q C [⋆ ]

Pic(C)

Lemma 14. Let Pice,m (C) ⊂ Pic(C) parameterize sheaves with Euler characteristic e which require m generators at p, and let Z Pe,m = h0 dχ Pice,m (C)

Then Pe,m = 0 for e ≤ −δ, and Pe,m −P−e,m = e·χ(Pic0,m (C)). Consequently there exist integers nh,m such that δ X X 2e nh,m (q −1 − q)2h−2 q Pe,m = e

h=0

Proof. The first statement follows from the vanishing of h0 in the specified range. For the second, use the duality involution M → M D := Hom C (M, ωC ) to identify Pice,m (C) and Pic−e,m (C). By Serre duality e(M D ) = −e(M) and h0 (M D ) = h0 (M) − e(M). By Lemma 12 and the local triviality of ωC , ˆ C,p , OC,p )) = m(M ˆ C,p ) = m(M) ˆC,p ) = m(Hom ˆ (M m(M D ) = m(Hom(M, ωC ) ⊗ O OC,p

Finally we may compute Pe,m − P−e,m =

Z

0

h dχ −

Pice,m (C)

Z

h0 dχ = e · χ(Pic0,m (C))

Pic−e,m (C)

This last integral is independent of e since twisting by line bundles does not affect m. The final statement of the lemma follows by linear algebra.  P Let nh (a2 ) = m (1 − a2 )m−1 nh,m so that Z δ X 2e 2 m−1 q (1 − a ) dχ = nh (a2 )(q −1 − q)2h−2 h=0

Pice (C)

9

As m is a constructible function it only assumes finitely many values on Pic0 ; identifying all the Pice by a choice of line bundle, we see m only assumes finitely many values in total. Thus the nh (a2 ) are polynomial. At a = 0, the nh (a2 ) recover the nh functions of [PT3, Appendix B]. Theorem 15. Let Cp be the germ of a plane curve; let µ be the Milnor number. Then there exist nh ∈ Z[a2 ] such that Z δ X 2l 2 m−1 µ µ 2 nh (a2 )(q −1 − q)2h+1−b dχ = a (a/q) (1 − q ) q (1 − a ) h=0

Cp [⋆ ]

Proof. It remains only to collect results and recall µ = 2δ + 1 − b [M].



Remark. Thus Conjecture 2 predicts a−µ (q −1 −q)b−1 P ∈ Z[a2 , (q −1 −q)2 ] has degree in (q −1 −q) at most µ. For unibranch singularities, we can also show that the degree in a2 is at most the multiplicity of the singularity. 5. T HE

CURVE

y k = xn

We consider now the singularity at the origin of y k = xn for k, n relatively prime. The complete local ring is C[[tk , tn ]], and the corresponding knot is the (k, n) torus knot. Jones has computed its HOMFLY polynomial. Theorem 16 (Jones). j k−1 2 jn+(k−1−j)(k−j)/2 Y (1 − q 2 )(a/q)(k−1)(n−1) X j (q ) P(Tk,n ) = (q 2i − a2 ) (−1) (1 − q 2k )(1 − a2 ) j=0 [j]! [k − 1 − j]! i=j+1−k

where [0]! = 1 and [r]! = (1 − q 2r )[r − 1]!

The Milnor number this singularity is µ = (k − 1)(n − 1). After rearranging the normalization factors, it suffices for us to prove: Z

[⋆ ]

Cp

j k−1 2 jn+(k−1−j)(k−j)/2 X Y 1 j (q ) (−1) (q 2i − a2 ) q (1 − a ) dχ = 2k 1 − q j=0 [j]! [k − 1 − j]! i=j+1−k 2l

2 m

To compute the left hand side we use a torus action. C∗ acts on C[[tk , tn ]] by scaling t. The action lifts to the Hilbert scheme, and preserves the functions l, m measuring length and number of generators. The integral with respect to Euler characteristic of an C∗ -equivariant function may always be computed on the fixed locus of the C∗ action since the remainder of the space will be fibred by C∗ and hence contribute zero to the Euler characteristic. Diagonalizing the C∗ action on a fixed ideal will yield monomial generators; conversely all the monomial ideals are fixed. There are countably many of these, and only finitely many with colength below any given bound. Therefore: Z X q 2 dimC O/J (1 − a2 )m(J) q 2l (1 − a2 )m dχ = [⋆ ]

Cp

J monomial

It remains to sum over the monomial ideals. We enumerate these in the usual way by matching them with staircases [Br, I]. Consider the map N × {0, . . . , k − 1} → monomials ∈ O (α, β) 7→ tαk+βn 10

It follows from the Chinese remainder theorem that this is a bijection. Monomial ideals are in 1-1 correspondence with certain staircases. Specifically, ideals are enumerated by sequences φ = φk−1 ≤ φk−2 ≤ . . . ≤ φ0 ≤ φk−1 + n via the correspondence φ ↔ {(α, β) | α > φβ } The number of generators of the ideal is the number of inequalities above which are strict. P The cardinality of the complement of the ideal is “the number of boxes under the staircase,” or φi .

Example 17. We give the staircase of (t21 , t23 , t24 ) monomials in the ideal. 15 19 23 27 31 10 14 18 22 26 5 9 13 17 21 0

4

8

12

16

⊂ C[[t4 , t5 ]]. Bold numbers correspond to

35 39 43 30 34 38 25 29 33 20 24 28

Example 18. The following staircase does not correspond to any ideal of C[[t4 , t5 ]], because 28 = 23 + 5. This occurs because the staircase does not descend quickly enough. 15 19 23 27 31 35 39 43 47 10 14 18 22 26 30 34 38 42 5 9 13 17 21 25 29 33 37 0 4 8 12 16 20 24 28 32 It remains to count the staircases with appropriate weights; this is done by the formula:

(16)

 k−1  1 Y ξq i 1 2 resξ=0 n+1 1 + (1 − a ) 1 − q 2k ξ 1 − ξq i i=0

Previously the φi described the rows of the staircase; the present formula describes the columns. The leading term (1 − q 2k )−1 accounts for the leading columns of full height k. The term i in the product corresponds to columns of height i. The number of different column heights is equal to the number of inequalities in φ. The residue enforces the condition that there should be exactly n columns of height less than k. We evaluate the residue by summing over the other singularities of the expression. These occur precisely at ξ = q −2j for j = 0, 1, . . . , k − 1. Note that 1 1 resz=1/w =− 1 − wz w in order to evaluate the residue: ! ! k−1 ! j−1 k−1 k−1 X Y Y Y 1 1 1 2(n+1)j −2j 2 i−j (17) q q 1−a q 1 − q 2k j=0 1 − q i−j 1 − q i−j i=0 i=j+1 i=0 It remains to collect signs and powers of q in order to prove:

Theorem 19. Let gcd(k, n) = 1. Let C be the curve cut out by y k = xn and let p be the origin; µ = (k − 1)(n − 1) is the Milnor number of this singularity, its link is the k, n torus knot, and Z µ 2 P( k,n torus knot ) = (a/q) (1 − q ) q 2l (1 − a2 )m−1 dχ Cp [⋆ ]

11

Corollary 20. Let C be a rational curve, away from a point p, and formally isomorphic at
smooth[b]! b k n p to Spec C[[x, y]]/(y = x ). Write c q2 for [c]![b−c]! . Then
Z k+n k q2 (1 − q 2 )2 q 2l dχ = k+n
1

C [⋆ ]

q2

Proof. We have proven Conjecture 2 in the case of the singularity in question, which implies Conjecture 1. Substituting in, we see Z j k−1 2 jn+(k−1−j)(k−j)/2 Y (1 − q 2 ) X j (q ) 2 2 2l (−1) q 2i (1 − q ) q dχ = 2k (1 − q ) j=0 [j]! [k − 1 − j]! i=j+1−k

C [⋆ ]

 k−1  (1 − q 2 ) X k − 1 = q j(j−1) (−q 2(n+1) )j j [k]! j=0 q2 Now we use the “Newton formula for Gaussian binomials”   s−1 s Y X k r(r−1) s t = (1 + q 2r t) q r q2 r=0 r=0 to deduce

(1 − q 2 )2

Z

C [⋆ ]

k−2

(1 − q 2 ) Y (1 − (q 2 )n+j+1) = q 2l dχ = [k]! j=0

k+n k q2
k+n 1 q2






Setting q = 1 recovers Beauville’s formula [B] for the Euler number of the compactified Jacobian. 6. H ILBERT

SCHEMES OF UNIBRANCH SINGULARITIES

Let C be the germ of a unibranch singularity, and fix a normalization O = OC ֒→ C[[t]]. Let ν : C[[t]] → N be the valuation taking a series to the degree of its lowest degree term. Lemma 10 shows that ν(f ) = dimC O/f O, so ν|O is independent of the choice of normalization. Let Γ = ν(O) – when appropriate we implicitly exclude 0 from the domain of ν – then Γ is a cofinite subset of N closed under addition.5 We employ the filtration Fk O = {f ∈ O|ν(f ) ≥ k}. Lemma 21. dim Fk O/Fk+1 O ≤ 1. If J an ideal of O, then ν(J) ⊂ N is a semigroup ideal: ν(J) + Γ ⊂ ν(J). Corollary 22. Let J be an ideal of O. Then dimC O/J = #ν(O) \ ν(J). P Proof. As dimC O/J is finite, dim O/J = dim Fn O/(Fn O ∩ J + Fn+1 O). This contributes 1 exactly when there is a ring element with valuation n, but no ideal element with valuation n; i.e., for n ∈ ν(O) \ ν(J).  For elements a0 , . . . , ak ∈ Γ, we denote the semigroup ideal they generate by (a0 , . . . , ak )Γ = {ai + γi |γi ∈ Γ} 5If not, then the fraction field of O would be some k((tr )) ( k((t)), but O and its normalization k[[t]] must share the same fraction field. For the theory of unibranch singularities, we refer to the book of Zariski and Tessier [ZT].

12

Corollary 23. Let J be an ideal of O. For f0 , . . . , fk ∈ J, (ν(f0 ), . . . , ν(fk ))Γ = ν(J) =⇒ (f0 , . . . , fk ) = J Proof. Let J ′ be the ideal generated by the fi . Since J ′ ⊂ J, surely ν(J ′ ) ⊂ ν(J). But ν(J ′ ) ⊃ (ν(f0 ), . . . , ν(fk ))Γ = ν(J) Thus dim J/J ′ = dim O/J ′ − dim O/J = #Γ \ J ′ − #Γ \ J = 0.



Remark. The converse is false. The ring O = k[[t4 , t6 + t7 ]] has semigroup h4, 6, 13i. Its maximal ideal M = (t4 , t6 + t7 ) has semigroup ideal ν(M) = (4, 6, 13)Γ. In fact, there is no ideal J with ν(J) = (4, 6)Γ: any such ideal contains t4 , t6 + t7 , hence (t6 + t7 )2 − (t4 )3 = 2t13 + t14 . We write Γ[⋆ ] for the set of semigroup ideals of Γ, and view ν as a constructible map ν : C [⋆ ] → Γ[⋆ ] which lets us define a stratification C [j] = ν −1 (j). Corollary 22 shows that this is a sub-stratification of the usual one by length: C [j] ⊂ C [#Γ\j] . Remark. Tessier [ZT] constructs a C∗ -equivariant deformation C → A1 whose generic fibre is C and whose special fibre is the not-necessarily-planar CΓ = Spec C[[Γ]]. The central fibre carries a ∗ natural C∗ action. Lifting the action to the Hilbert scheme, (CΓ [⋆ ] )C = Γ[⋆ ] . The map ν amounts to taking the t → 0 limit of the C∗ action on the relative Hilbert scheme of C/A1 . Definition 24. For g ∈ Γ, let τg ∈ O ⊂ C[[t]] be the unique element of the form X τg = tg + ci ti gai \ (a0 , . . . , ak )Γ , and let Va0 ,...,ak = Spec C[Sλ,i∈Σλ ]. Define fλ ∈ O ⊗C C[Sλ,i∈Σλ ] by X τi Sλ,i fλ = τaλ + i∈Σλ

and consider the ideal J = (f0 , . . . , fk ) ⊂ O[Sλ,i∈Σλ ]. Let

Ua0 ,...,ak = {s ∈ Va0 ,...,ak | dim O/Js = #Γ \ (a0 , . . . , ak )Γ } As we always have (a0 , . . . , ak )Γ ⊂ ν(J |s ), this is the locus where ν(J |s ) = (a0 , . . . , ak )Γ . Take it with the reduced induced scheme structure. As the Hilbert polynomial is constant and the base is reduced, J is flat over Ua0 ,...,ak . Thus we get a map Ψa0 ,...,ak : Ua0 ,...,ak → C [(a0 ,...,ak )Γ ] ⊂ C [⋆ ] Theorem 25. Ψa0 ,...,ak : Ua0 ,...,ak → C [(a0 ,...,ak )Γ ] is bijective. [j] Proof. Let j = (a0 , . . . , ak )Γ . Consider a closed point (sλ,i∈Σ P λ ) in the preimage of C . This corresponds to an ideal J = (f0 , . . . , fk ) where fi = τaλ + i∈Σ J is also the Pλ τi sλ,i . Suppose ′ ′ ′ ideal corresponding to (sλ,i∈Σλ ), hence has generators fλ = τaλ + i∈Σλ τi sλ,i . Now fλ − fλ′ ∈ J, but on the other hand ν(fλ − fλ′ ) ∈ Γ \ j unless fλ = fλ′ . Thus the map is injective. For surjectivity, fix an ideal J with ν(J) =P j. Choose lifts of fλ ∈ J of the aλ . A generating set will still generate if we modify fλ → ufλ + ν6=λ vfν for invertible u and arbitrary v. Iteratively removing terms of the form tn for n ∈ j \ λ from fλ converges to yield generators of J of the form required by Definition 24. 

13

Remark. As defined, the Va0 ,...,ak depend on the choice of generators of the semigroup ideal (a0 , . . . , ak )Γ . However, a semigroup ideal has a unique minimal generating set; henceforth if we write Vj to mean that the minimal generating set of the semigroup ideal is chosen. On the other hand, while it may likewise seem that Ua0 ,...,ak depends on the choice of generators, Theorem 25 implies that all choices yield spaces which biject onto C [j] . Caution. The function giving the number of generators need not be constant on the Uj . 7. T HE

SINGULARITY WITH SEMIGROUP

h4, 6, 13i

We consider now the ring O = C[[t4 , t6 + t7 ]] and the singularity C = Spec O. Let us calculate the semigroup. As (t6 + t7 )2 − (t4 )3 = t13 (2 + t) we see 4, 6, 13 ∈ ν(O). Suppose there is P (x, y) ∈ C[[x, y]] such that P (t) = P (t4 , t6 + t7 ) has leading term t15 . Then certainly P (x, y) must have two monomials, xa y b and xc y d , such that 4a + 6b = 4c + 6d < 15; moreover their leading terms must cancel when evaluated at x = t4 , y = t6 + t7 . By inspection, the first condition is only satisfied for 4a + 6b = 12, but in this case we have already seen that the leading term of P (t) is t13 . So 15 ∈ / ν(O), and the semigroup is Γ = h4, 6, 13i = {0, 4, 6, 8, 10, 12, 13, 14, 16, 17, . . .} In fact, Zariski has shown that this is the only plane singularity with this semigroup [ZT]. The link of this singularity is the (2,13) cable of the (2,3) torus knot [EN]. Its HOMFLY polynomial, as calculated by computer, is − + − +

a22 a20 a18 a16

(3 + 4z 2 + z 4 ) (20 + 70z 2 + 84z 4 + 45z 6 + 11z 8 + z 10 ) (39 + 220z 2 + 468z 4 + 496z 6 + 286z 8 + 91z 10 + 15z 12 + z 14 ) (23 + 179z 2 + 540z 4 + 836z 6 + 726z 8 + 365z 10 + 105z 12 + 16z 14 + z 16 )

where z = q −q −1 . According to Conjecture 1, the coefficient of z 2h in the bottom row above is the number nh of Pandharipande and Thomas [PT3, Appendix 2]. In particular, the Euler characteristic of the Jacobian factor of this singularity should be n0 = 23. This was previously calculated by Piontkowski [Pi] using similar methods.6 We turn now to the calculation of the integral in Conjecture 2 by using the stratification of Section 6. Moreover we group the semigroup ideals which are isomorphic as Γ-modules, i.e., differ merely by a shift. Let Mod(Γ) denote the set of Γ submodules of N containing zero. Using Theorem 25 and Corollary 11, Z Z X X 2l 2 m−1 −(8−#N\∆) 2i (18) q (1 − a ) dχ = q q (1 − a2 )m−1 dχ C [⋆ ]

i+∆⊂Γ

∆∈Mod(Γ)

Ui+∆

The combinatorial data required to compute the right hand side is tabulated in Figure 8. We compute the rightmost integral by determining the spaces Ui+∆ , together with their stratifications by the number of generators. 6

Indeed, Piontkowski also determines the Euler characteristic of the Jacobian factor for all singularities with semigroups h4, 2q, si, h6, 8, si, and h6, 10, si. He does so by constructing a stratification of the Jacobian factor by affine spaces, and suggests that it is unlikely that any other singularities will admit such a stratification. We remark that his list exhausts the algebraic (2, n) cables of (2, p), (3, 4) and (3, 5) torus knots, or in other words, the (2, n) cables of knots arisisng from simple singularities. In [ORS] we will check the conjecture for these singularities. 14

Lemma 26. Let Γ = h4, 6, 13i, and consider ∆ ∈ Mod(Γ). Let 0 = α0 < α1 < . . . be its minimal Pset of generators. Choose arbitrary f0 , f1 , . . . ∈ k[[t]] with degrees α0 , α1 , . . .. Then if η = ν( fi φi ) ∈ / ∆ for φi ∈ O, then: • α1 = 2 and 1, 3 ∈ /∆ • η ∈ {7, 9, 11, 15} • ν(f0 φ0 ) = ν(f1 φ1 ) = min{ν(fi φi )} P Proof. Assume there exist φ ∈ O such that η = ν( fi φi ) ∈ / ∆. In particular, we must have i P ν( fi φi ) > min ν(fi φi ), thus the lowest degree terms must cancel, thus there must be at least two of them. Say they are fj and fk ; let αj < αk . We have η > αk + ν(φk ) = αj + ν(φj )

We cannot have ν(φk ) = 0 since αk is a necessary generator; thus ν(φk ) ≥ 4. Since ν(φj ) > ν(φk ) we also have ν(φj ) ≥ 6. As η > αj +ν(φj ) ≥ 6 and η ∈ / ∆, we must have η ∈ {7, 9, 11, 15}. Since η ∈ / ∆, no odd number less than η − 2 can be in ∆. This implies that 1, 3 ∈ / ∆ and aj , ak are even. This can only happen if aj = 0 and ak = 2.  Lemma 27. Given two series f0 = 1 + a1 t + a3 t3 + . . . f1 = t2 (1 + b1 t + b3 t3 + . . .) we see that degt ((t6 + t7 )f0 − t4 f1 ) ≥ 7 with equality unless b1 − a1 = 1 degt ((t6 + t7 )f1 − t8 f0 ) ≥ 9 with equality unless a1 − b1 = 1 The equations cannot hold simultaneously, so at least one of the series has the specified degree. Corollary 28. Let 0 ∈ ∆ ⊂ N be a Γ-module with minimal generators α0 , α1 , . . .. Choose lifts f0 , f1 , . . . ∈ C[[t]]. Let ∆′ = ν((f0 , f1 , . . .)). Then if ∆ 6= ∆′ , then ∆ → ∆′ appears on the following list. • (0, 2) → (0, 2, 7), (0, 2, 9), (0, 2, 7, 9) • (0, 2, 5) → (0, 2, 5, 7) • (0, 2, 7) → (0, 2, 7, 9) • (0, 2, 9) → (0, 2, 7, 9), (0, 2, 9, 11) • (0, 2, 11) → (0, 2, 7), (0, 2, 7, 9), (0, 2, 9, 11) For all modules Φ not occuring on the list, Z (1 − a2 )m−1 dχ = (1 − a2 )m(Φ)−1 Ui+Φ

where m(Φ) is the number of generators of Φ as a Γ-module. Moreover, if Φ = (0, 2) or (0, 2, 11), then Ui+Φ = ∅, so Z (1 − a2 )m−1 dχ = 0

Ui+Φ

15

Proof. All Γ modules are listed in Figure 8. Checking the criterion of Lemma 26 on each of them yields the list of possible ∆. The list of possible ∆′ comes from the criterion of Lemma 27: if 0 and 2 are in ∆, then 7 or 9 is in ∆′ . If a semigroup module M never occurs as one of the ∆ above, Theorem 25 implies that Ui+M is an affine space. If M never occurs as one of the ∆′ , then Theorem 25 implies that ν(I) = i + M =⇒ m(I) = m(M). The final statement of the corollary is immediate from Lemma 27.  We proceed to analyse the remaining modules. Suppose ∆ ∈ Mod(Γ), 0, 2 ∈ ∆, and 1, 3 ∈ / ∆. Fix i such that i + ∆ ⊂ Γ, and let Vi+∆ be the affine space of Definition 24. Vi+∆ has coordinate functions a, b giving respectively the coefficient of xi+1 of the generator of degree i, and the coefficient of xi+3 in the generator of degree i + 2. Let I be an ideal corresponding to some point in Vi+∆ such that a = a(I) and b = b(I). Lemma 27 implies that i + 7 ∈ ν(I) unless b − a = 1, and i + 9 ∈ ν(I) unless a − b = 1. In fact, b − a is identically 1 if and only if i + 7 ∈ / Γ, identically −1 if and only if i + 9 ∈ / Γ, and otherwise may assume any value. This may be seen from the explicit form of a general element of O: X c0 + c4 t4 + c6 (t6 + t7 ) + c8 t8 + c10 (t10 + t11 ) + c12 t12 + c13 t13 + c14 (t14 + t15 ) + cn tn n≥16

We will now determine the Uj of Definition 24 by computing their complements in the affine spaces Vj . Recall that the space Vj depended on a choice of generators of j; we use the set of generators indicated (which is always the minimal set of generators). For I an ideal of O, we say its type is the semigroup ideal ν(I) ⊂ Γ. • The complement of Ui+(0,2,5) inside Vi+(0,2,5) will be the ideals whose type is i+ (0, 2, 5, 7). As i + 9 ∈ Γ since i + 5 ∈ Γ, the i + 7 appears in the complement of a hyperplane if at all. Thus UΓ has Euler characteristic 1. On the other hand, we see in Corollary 28 that an ideal of type i + (0, 2, 5) always has three generators. Thus Z (1 − a2 )m−1 dχ = (1 − a2 )2 Ui+(0,2,5)

• The complement of Ui+(0,2,7) in Vi+(0,2,7) consists of semigroup ideals whose type is i + (0, 2, 7, 9). Since i+7 is in the semigroup ideal, i+9 fails to be in the semigroup ideal either on a hyperplane or in all of Ui+(0,2,7) . Moreover when i + 9 fails to be in the semigroup ideal, by Lemma 27, the generators of degree i, i + 2 generate the ideal. Thus Z (1 − a2 )m−1 dχ = (1 − a2 ) Ui+(0,2,5)

• Ui+(0,2,5,7) = Vi+(0,2,5,7) . Since i + 9 is in the semigroup ideal, the space on which the generator of degree seven is not needed is the complement of a hyperplane and thus has Euler characteristic zero. Thus: Z (1 − a2 )m−1 dχ = (1 − a2 )3 Ui+(0,2,5,7)

• Ui+(0,2,7,9) = Vi+(0,2,7,9) . Since 7 and 9 are both in the semigroup ideal, the generator of degree i + 7 is unnecessary in the complement of a hyperplane, and the generator of degree 16

i + 9 is unnecessary in the complement of a parallel hyperplane. Z (1 − a2 )m−1 dχ = 2(1 − a2 )2 − (1 − a2 ) Ui+(0,2,7,9)

• The complement of Ui+(0,2,9) in Vi+(0,2,9) consists of the locus where the type is (0, 2, 7, 9) or (0, 2, 9, 11). We already understand that the first happens, if at all, on the the complement ′ of a hyperplane. Let Ui+0,2,9 be the affine space on which the type is not (0, 2, 7, 9). Here the generators may be written: f = ti + ati+1 + a3 ti+3 + a5 ti+5 + a7 ti+7 + a11 ti+11 g = ti+2 + (a + 1)ti+3 + b3 ti+5 + b5 ti+7 + b9 ti+11 h = ti+9 + cti+11 Writing x = t4 and y = t6 + t7 , we see that (x2 + xy(a + 1))f − yg + 2h = ((a + 1)2 + 2c + a3 − b3 )t11 + O(t12 ) ′′ Now let Ui+(0,2,9) be the locus where the above coefficient of t11 vanishes; since it is given ′′ as c = ((b3 − a3 ) + (a + 1)2 )/2, it is isomorphic to affine space. Restricting to Ui+(0,2,9) , 12 one checks that the only term of degree 11, modulo t , is

(2x − (a2 − a3 + b3 )y)g + (2(ax2 + a3 xy − y) + (a2 − a3 + b3 )(x2 + xy(a + 1)))f and the coefficient of t11 is C(a, a3 , a5 , b3 , b5 ) = 2b5 − 2a5 + P (a, a3 , b3 )), where P is ′′ where C vanishes. If some polynomial. Evidently Ui+(0,2,9) is the locus inside Ui+(0,2,9) either i + 5 or i + 7 is in Γ, then C vanishes on a hypersurface isomorphic to affine space. In fact, by inspection of Γ, one sees that whenever i + (0, 2, 9, 11) ⊂ Γ, either i + 5 or i + 7 is in Γ. In the case i + (0, 2, 9, 11) 6⊂ Γ and hence i + 11 ∈ / Γ, then C vanishes ′′ identically, Ui+(0,2,9) = Ui+(0,2,9) , which we already knew was an affine space. In any event Ui+(0,2,9) has Euler characteristic 1. Finally by Lemma 27 the generator of degree i + 9 is superfluous, so: Z (1 − a2 )m−1 dχ = (1 − a2 ) Ui+(0,2,9)

• The complement of Ui+(0,2,9,11) inside Vi+(0,2,9,11) is the locus where thes semigroup type is i + (0, 2, 7, 9). We have seen that the i + 7 appears in the complement of a hyperplane, if at all. Thus Ui+(0,2,9,11) is isomorphic to affine space. We have also seen that the generator of degree i + 9 is always superfluous; by the argument given for (0, 2, 9), the generator of degree i + 11 is superfluous in the complement of an affine space. Thus: Z (1 − a2 )m−1 dχ = (1 − a2 )2 Ui+(0,2,9,11)

This completes the determination of the integrals appearing in Equation (18). Summing the contributions yields complete agreement with the HOMFLY polynomial.

17

8. F URTHER

DIRECTIONS

Towards a proof. We see two possible approaches. First, one might try and emulate the proof of Campillo, Delgado, and Gusein-Zade [CDG]. Roughly speaking, they provide a formula for computing the RHS of Equation (13) from a resolution diagram of a singularity, and match it to the Eisenbud-Neumann [EN] formula computing the Alexander polynomial from the same. Tt may be possible to describe how the Hilbert scheme transforms under blowups – see, e.g., [NY] – but one would still need an analogue of the Eisenbud-Neumann formula for the HOMFLY polynomial. Second, one may try to establish the skein relation. Randomly changing the crossings in the link of a singularity is not likely to produce the link of another singularity. However, taking the singularity to be given locally as the zero locus of f (x, y), consider a two-parameter deformation f (x, y) + ǫ1 g(x, y) + ǫ2 h(x, y). Suppose that somewhere in this deformation, the singularity splits off a cusp, whereas generically it splits off two nodes. Now imagine starting with a sphere far away from the singular locus, and shrinking it past one node. On the knot diagram, this corresponds to smoothing a crossing. Shrinking past both nodes switches this crossing. Thus we have all the terms in the skein relation. However, we know neither whether such deformations always exist, nor how they affect the Hilbert schemes. Other knot invariants. Conjecture 2 relates the HOMFLY polynomial to the space of ideals of the ring of functions on a singular curve. At a = −1, this specializes to a relation between the Alexander polynomial and the space of principal ideals; here it has been proven by Campillo, Delgado, and Gusein-Zade [CDG]. The HOMFLY polynomial admits other well-known specializations: most famously, at a = q 2 it recovers the Jones polynomial. More generally, setting a = q n recovers the quantum sln invariant. At n = 1, the invariant is trivial and thus Conjecture 2 predicts Z q 2l (1 − q 2 )m dχ = 1 C [⋆ ]

We do not have a proof of this identity, which numerical experiments suggest is not restricted to planar singularities. More generally, it would be very interesting know the geometric meaning of the substitution a = q n . In fact, there is a polynomial invariant associated to every representation of every simple Lie algebra; the HOMFLY polynomial encodes only those invariants arising from the fundamental representation of sln . Do the others also arise as generating functions of Euler characteristics of moduli spaces? Possible connections with physics. As we remarked in the introduction, the generating function of Euler characteristics of Hilbert schemes of surfaces has some physical significance. The generating function of Euler characteristics of singular curves appears in the work of Pandharipande and Thomas [PT3] as a way to define the Gopakumar-Vafa [GV] numbers for curves on a K3 surface. The curve counting theory they study is a complex analogue of Chern-Simons gauge theory. On the other hand, Witten [W] has shown that the HOMFLY polynomial admits an interpretation in terms of correlation functions in the original Chern-Simons gauge theory on the three-sphere. This has since been reformulated in the language of open string theory [OV], where the HOMFLY polynomial is recovered from the Euler characteristics of certain moduli spaces of open strings. Perhaps some physical duality relates this open-string picture to the algebraic one described here.

18

R EFERENCES [AGV] V. Arnold, S. Gusein-Zade, A. Varchenko, Singularities of Differentiable Maps, Vol. I & II, Birkhauser 1988. [B] Arnaud Beauville, Counting rational curves on K3 surfaces, Duke Math. J. 97.1, pp. 99-108 (1999). [Br] J. Brianc¸on, Description de Hilbn C{x, y}, Invent. Math. 41, pp. 45-89 (1977). [vB] J. M. von Buskirk, Positive knots have positive Conway polynomials in Lecture Notes in Mathematics 1144 pp. 146-159, Springer (1985). [CDG] A. Campillo, F. Delgado, and S. M. Gusein-Zade, The Alexander polynomial of a plane curve singularity via the ring of functions on it, Duke Math J. 118.1, pp. 125-156 (2003). [E] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. AMS (1980). [EN] D. Eisenbud and W. Neumann, Three-dimensional link theory and invariants of plane curve singularities, Ann. of Math. Studies 110, Princeton, 1985. [HOMFLY] P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12, p. 239 (2002). [Gt] L. G¨ottsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 pp. 193-207 (1990). [Gr] I. Grojnowski, Instantons and affine algebras I: The Hilbert scheme and vertex operators, Math. Res. Lett. 3, pp. 275-291 (1996). [G] A. Grothendieck, Techniques de construction et th´eor`ems d’existence en g´eom´etrie alg´ebraique. IV. Les sch´emas de Hilbert, S´eminare Bourbaki, Vol. 6, Exp. 221, Soc. Math. France, Paris, pp. 249-276 (1995). [GV] R. Gopakumar and C. Vafa, M-theory and topological strings II, hep-th/9812127. [I] A. Iarrobino. Punctual Hilbert schemes. Mem. Am. Math. Soc. 188 (1977). [J] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math 126, pp. 335-388 (1987). [KKV] S. Katz, A. Klemm, and C. Vafa, M-theory, topological strings, and spinning black holes, Adv. Theor. Math. Phys. 3, pp. 1445-1537 (1999). [KR] M. Khovanov, L. Rozansky, Matrix factorizations and link homology, I. [math.QA/0401268] and II. [math.QA/0505056]. [KY] T. Kawai and K. Yoshioka, String partition functions and infinite products, Adv. Theor. Math. Phys. 4, pp. 397-485 (2000). [M] J. Milnor, Singular points of complex hypersurfaces, Princeton University Press 1968. [N] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math 145, pp. 379-399 (1997). [NY] H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blow-up. II. Wall-crossing and Betti numbers formula, [arXiv:0806.0463]. [ORS] A. Oblomkov, J. Rasmussen, and V. Shende, The Hilbert scheme of a singular curve and the HOMFLY homology of its link, to appear. [OV] H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577, p. 419 (2000) [arXiv:hep-th/9912123]. [Pi] J. Piontkowski. Topology of the compactified Jacobians of singular curves, Mathematische Zeitschrift bf 255.1, pp. 195-226 (2007) [P] R. Pandharipande, Hilbert schemes of singular curves, web notes. [PT3] R. Pandharipande and R. Thomas, Stable pairs and BPS invariants [arXiv:0711.3899]. [VW] C. Vafa and E. Witten, A strong coupling test of S-duality, Nuclear Phys. B 431, pp 3-77 (1994). [W] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121, p. 351 (1989). [W2] E. Witten, Chern-Simons Gauge Theory As A String Theory, Prog. Math. 133, p. 637 (1995) [arXiv:hep-th/9207094]. [Z2] O. Zariski, Studies in Equisingularity. II. Equisingularity in codimension 1 (and characteristic zero), Amer. J. Math. 87, pp. 972-1006 (1965). [ZT] O. Zariski and B. Tessier, Le probl`eme des modules pour les branches planes, Hermann, Paris, 1986.

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∆ ∈ Mod(Γ) 8 − #N \ ∆ i|i + ∆ ⊂ Γ (0) 0 0, 4, 6, 8, 10, 12, 13, 14, 16+ (0, 1) 6 12, 13, 16+ (0, 3) 5 10, 13, 14, 16+ (0, 5) 4 8, 12, 13, 14, 16+ (0, 7) 3 6, 10, 12, 13, 14, 16+ (0, 9) 2 4, 8, 10, 12, 13, 14, 16+ (0, 11) 2 6, 8, 10, 12, 13, 14, 16+ (0, 15) 1 4, 6, 8, 10, 12, 13, 14, 16+ (0, 2) 2 4, 6, 8, 10, 12, 14, 16+ (0, 1, 3) 7 13, 16+ (0, 3, 5) 6 13, 14, 16+ (0, 5, 7) 5 12, 13, 14, 16+ (0, 7, 9) 4 10, 12, 13, 14, 16+ (0, 9, 11) 3 8, 10, 12, 13, 14, 16+ (0, 1, 2) 7 12, 16+ (0, 2, 3) 6 10, 14, 16+ (0, 2, 5) 5 8, 12, 14, 16+ (0, 2, 7) 4 6, 10, 12, 14, 16+ (0, 2, 9) 3 4, 8, 10, 12, 14, 16+ (0, 2, 11) 3 6, 8, 10, 12, 14, 16+ (0, 1, 2, 3) 8 16+ (0, 2, 3, 5) 7 14, 16+ (0, 2, 5, 7) 6 12, 14, 16+ (0, 2, 7, 9) 5 10, 12, 14, 16+ (0, 2, 9, 11) 4 8, 10, 12, 14, 16+ F IGURE 1. The combinatorial data required by Equation (18).

Alexei Oblomkov Department of Mathematics University of Massachusetts, Amherst Amherst, MA 01003 [email protected] Vivek Shende Department of Mathematics Princeton University Princeton NJ, 08540 [email protected]

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