SEGRE CLASSES AND HILBERT SCHEMES OF POINTS
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1 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS A. MARIAN, D. OPREA, AND R. PANDHARIPANDE Abstract. We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a K3 surface X. We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on X. The resulting recursions are then solved explicitly. The formula proves the K-trivial case of a conjecture of M. Lehn from The relations determining the Segre classes fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes on surfaces, we obtain new systems of relations among tautological classes on moduli spaces of surfaces and their relative Hilbert schemes of points. For the moduli of K3 sufaces, we produce relations intertwining the κ classes and the Noether-Lefschetz loci. Conjectures are proposed. 0. Introduction 0.1. Segre classes. Let S, H) be a pair consisting of a nonsingular projective surface S and a line bundle H S. The degree of the pair S,H) is defined via the intersection product on S, H H = H 2 Z. The Hilbert scheme of points S [n] carries a tautological rank n vector bundle H [n] whose fiber over ζ S [n] is given by ζ H 0 H O ζ ). The top Segre class N S,H,n = s 2n H [n] ) S [n] appeared first in the algebraic study of Donaldson invariants via the moduli space of rank 2 bundles on S [25]. Such Segre classes play a basic role in the Donaldson-Thomas counting of sheaves often entering via the obstruction theory). A classical interpretation of N S,H,n is also available. If H is a linear system of dimension 3n 2 which induces a map S P 3n 2, N S,H,n counts the n-chords of dimension n 2 to the image of S. S Date: July
2 2 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE The main result of the paper is the calculation of the top Segre classes for all pairs X,H) in the K3 case. Theorem 1. If X,H) is a nonsingular K3 surface of degree 2l, then ) l 2n+2 s 2n H [n] ) = 2 n. X [n] n 0.2. Lehn s conjecture. Let S be a nonsingular projective surface. The Segre class N S,H,n can be expressed as a polynomial of degree n in the four variables H 2, H K S, K 2 S, c 2S), see [23] for a proof. Furthermore, the form ) 1) N S,H,n z n = exp H 2 A 1 z)+h K S ) A 2 z)+ks 2 A 3 z)+c 2 S) A 4 z) in terms of four universal power series A 1,A 2,A 3,A 4 was proven in [5]. The formulas for the four power series were explicitly conjectured by M. Lehn in Conjecture 1 Lehn [12]). We have 2) N S,H,n z n = 1 w)a 1 2w) b 1 6w +6w 2 ) c for the change of variable and constants z = w1 w)1 2w)4 1 6w +6w 2 ) 3 a = H K S 2K 2 S, b = H K S) 2 +3χO S ), c = 1 2 HH K S)+χO S ). As usual in the study of the Hilbert scheme of points, Theorem 1 determines two of the power series in 1). Specifically, Theorem 1 implies 3) A t1+t)2 ) = 1 2 log1+t), 4) A t1+t)2 ) = 1 8 log1+t) 1 24 log1+3t). The evaluation of A 1 and A 4 proves Lehn s conjecture for all surfaces with numerically trivial canonical bundle. Corollary 1. If A,H) is an abelian or bielliptic surface of degree 2l, then ) s 2n H [n] ) = 2n l l 2n 1. A [n] n n 1
3 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 3 Corollary 2. If E,H) is an Enriques surface of degree 2l, then ) 2 ) 2l 2n+2 N E,H,n z n = 2 n z n. n 0.3. Strategy of the proof. The intersection theory of the Hilbert scheme of points can be approached via the inductive recursions set up in [5] or via the Nakajima calculus [12, 18]. By these methods, the integration of tautological classes is reduced to a combinatorial problem. Another strategy is to prove an equivariant version of Lehn s conjecture for the Hilbert scheme of points of C 2 via appropriately weighted sums over partitions. However, we do not know how to prove Theorem 1 along these lines. 1 Let X be a nonsingular projective K3 surface. We consider integrals over the Quot scheme Q H,χ C 2 ) parametrizing quotients C 2 O X F 0 where F is a rank 0 coherent sheaf satisfying c 1 F) = H and χf) = χ. The Quot scheme admits a reduced virtual class, and the integrals γ 0k [Q H,χ C 2 red )] vanish for all k > 0 and for all choices of cohomology classes γ. Virtual localization [7] with respect to a C action applied to the above integrals yields linear recursions between the expressions X [n] c n i H [n] )s n+i H [n] ). ThelinearrecursionsaretrivialforallbutfinitelymanyvaluesofH 2 = 2l. Thenontrivial recursions can be solved to show that the top Segre class vanishes for the values 2n 2 l 3n 3. These vanishings determine the intersections up to an ambiguity given by the leading term of a polynomial which we can calculate explicitly Plan of the paper. Section 1 concerns localization on Quot schemes. Foundational aspects of the virtual classes of Quot schemes on surfaces S are discussed in Section 1.1. The virtual localization formula for the C action on the Quot schemes of K3 surfaces is presented in Section 1.2. Explicit localization relations are derived in Section 1.3. Theorem 1 is proven in Section 2 by solving the recursion relations of Section 1.3. In Section 3, the connections between Theorem 1 and Lehn s conjecture are explained and 1 The parallel problem in dimension 1, the calculation of the Segre classes of tautological bundles over Hilbert schemes of points of nonsingular curves, has been solved in [3, 11, 26].
4 4 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE Corollaries 1 and 2 are proven). An application to elliptically fibered surfaces is given in Corollary 3 of Section 3. In Section 4, we explain how the relations used to prove Theorem 1 fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes over surfaces, we obtain new systems of relations among tautological classes on moduli spaces of surfaces and their relative Hilbert schemes of points. For the moduli of K3 sufaces, we produce relations intertwining the κ classes and the Noether-Lefschetz loci. Conjectures are proposed Acknowledgements. We thank N. Bergeron, G. Farkas, M. Lehn, D. Maulik, G. Oberdieck, and Q. Yin for several discussions related to tautological classes, Quot schemes, and the moduli space of K3 surfaces. The study of the relations presented here was undertaken during a visit of A.M. and D.O. in the spring of 2015 to the Institute for Theoretical Sciences at ETH Zürich and supported in part by SwissMap). A.M. was supported by the NSF through grant DMS D. O. was supported by the Sloan Foundation and the NSF through grants DMS and DMS R.P. was supported by the Swiss National Science Foundation and the European Research Council through grants SNF and ERC-2012-AdG MCSK. R.P was also supported by SwissMap and the Einstein Stiftung in Berlin. 1. Localization on the Quot scheme 1.1. Geometric setup. Let S be a nonsingular projective surface equipped with a divisor class H. We consider the Quot scheme Q H,χ C r ) parametrizing short exact sequences 0 E C r O S F 0 where F is a rank 0 coherent sheaf satisfying c 1 F) = H and χf) = χ. With the exception of the Hilbert scheme of points S [n] = Q 0,χ C 1 ) the intersection theory of Quot schemes over surfaces has not been extensively studied. Rank 1 calculation can be found in [4], and higher rank calculations over del Pezzo surfaces have been considered in [22]. In comparison, the intersection theory of the Quot scheme of a curve may be pursued in a virtual sense for a fixed curve [13] or by letting the curve vary via the moduli space of stable quotients [14]. The relations in the tautological ring of the moduli of curves [9, 20, 21] via virtual localization on the moduli stable quotients are parallel to the relations we introduce in Section 4.
5 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 5 Fundamental to our study is the following result which we will use here only in the r = 2 case). Lemma 1. The Quot scheme Q H,χ C r ) admits a canonical perfect obstruction theory of virtual dimension rχ+h 2. Proof. Sincethedetails aresimilartothecurvecase[13], weonlydiscussthemainpoints. The obstruction theory of the Quot scheme is governed by the groups Ext i E,F). We claim the vanishing Ext 2 E,F) = Ext 0 F,E K S ) = 0. Indeed, since E is a subsheaf of C r O S, the latter group injects Ext 0 F,E K S ) Ext 0 F,C r K S ) = 0, where the last vanishing follows since F is torsion. As a consequence, the difference 5) Ext 0 E,F) Ext 1 E,F) = χe,f) = rχ+h 2 is constant. Since the higher obstructions vanish, the moduli space Q H,χ C r ) carries a virtual fundamental class of dimension 5). Let X,H) be a primitively polarized K3 surface of degree 2l and Picard rank 1. Let H 2 = 2l and n = χ+l. In the K3 case, we show the Quot scheme admits a reduced virtual fundamental class. Lemma 2. For a K3 surface X,H) there is a natural surjective map Ext 1 E,F) C, and a reduced virtual fundamental class [Q H,χ C r )] red of dimension rχ+2l+1. Proof. The argument is standard. Indeed, the defining short exact sequence induces a natural morphism 0 E C r O X F 0 Ext 1 E,F) Ext 2 F,F) Trace H 2 O X ) = C. To prove surjectivity of the composition, it suffices to show Ext 1 E,F) Ext 2 F,F) is surjective, since the trace is surjective. The cokernel of the map is identified with Ext 2 C r,f) = H 2 F) C r
6 6 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE which vanishes since F has 1-dimensional support. The reduced virtual dimension equals χe,f)+1 = χc r,f) χf,f)+1 = rχ+h 2 +1 = rχ+2l+1, where the summand 1 on the left is due to the reduction. In the r = 2 case, the virtual dimension formula specializes to 2χ+l)+1 = 2n Virtual localization. We continue to work with a primitively polarized K3 surface X,H) of degree 2l and Picard rank 1. We study the virtual intersection theory of the Quot scheme Q H,χ C 2 ) via the virtual localization formula [7]. To this end, consider the diagonal C action, C 2 = C[0]+C[1], with weights 0 and 1 on the two summands. A torus action on Q H,χ C 2 ) is canonically induced. The torus fixed quotients come from kernels which split into rank 1 factors 0 E = E 1 E 2 O X O X F = F 1 +F 2 0. Since c 1 F) = H, we have c 1 E 1 ) c 1 E 2 ) = H. The two sections show that E1 and E 2 are effective line bundles. Since the curve class H is irreducible, the kernel must split as E 1 = I Z, E 2 = I W O X C) or E 1 = I Z O X C), E 2 = I W, where C is a curve in the linear series H and Z,W X are 0-dimensional subschemes of lengths z and w respectively. The condition χf) = χ is equivalent to z +w = l+χ = n. For each value of z and w, we obtain two distinct fixed loci, both isomorphic to the product X [z] X [w] P where the projective space P = H denotes the linear series in which the curve C varies. We now turn to the virtual class and the virtual normal bundle of the fixed loci. Except in the two cases z,w) = 0,n) and E 1 E 2 = O X I W O X C),
7 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 7 z,w) = n,0) and E 1 E 2 = I Z O X C) O X, the reduced obstruction bundle has an additional trivial summand which forces the localization contribution to vanish. The non-reduced obstruction bundle is obtained by considering the fixed part of Ext 1 E,F) which splits as Ext 1 E 1,F 1 )+Ext 1 E 2,F 2 ). By the proof of Lemma 2, if F 1 and F 2 are both non-zero, we have two surjective maps Ext 1 E 1,F 1 ) C, Ext 1 E 2,F 2 ) C. Since the reduced virtual class is obtained by reducing only one trivial factor, a trivial factor still remains. We now analyze the two surviving fixed components F + and F associated to the splittings E 1 E 2 = O X I W O X C) and E 1 E 2 = I Z O X C) O X respectively. The isomorphisms F + F X [n] P are immediate. Let L be the hyperplane class on P, L = O P 1). The line bundle associated to the universal curve C X P of the linear series H is OC) = H L. On the universal surface of the fixed locus F + X F +, the universal subsheaves are E 1 = O, E 2 = I W O C) = I W H 1 L 1. Tofindthereducedvirtual class of F +, we studythefixedpart of thereduced obstruction theory: Tan Obs) fix = Ext E 2,F 2 )+C = Ext I W H 1 L 1,C I W H 1 L 1 )+C = Ext I W H 1 L 1,C) Ext I W,I W )+C = Ext I W H 1,C) L Ext I W,I W )+C = H 0 X,H) L Ext O W H 1,C) L Ext I W,I W )+C. The latter expression simplifies using the following four observations:
8 8 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE i) Ext 1 I W,I W ) Ext 0 I W,O W ) is the tangent space to X [n] at W, ii) Ext 2 I W,I W ) = Ext 0 I W,I W ) = C by Serre duality, iii) Ext 0 O W H 1,C) = Ext 1 O W H 1,C) = 0 since W is of dimension 0, iv) Ext 2 O W H 1,C) = H 0 H 1 O W ) by Serre duality. Applying i-iv), we find Tan Obs) fix = C l+2 L L H 1 ) [n]) +TanX [n] C = Tan P L H 1 ) [n]) +TanX [n] where we have also used the Euler sequence 0 O C l+2 L Tan P 0. Therefore, over the fixed locus F +, we have the obstruction bundle Obs = L H 1 ) [n] ). Hence, the virtual class of the fixed locus F + is [ F + ] vir = el H 1 ) [n] ) ). The analysis of the fix part of the reduced obstruction theory for F is identical. We find the virtual class of the fixed locus is [ F ] vir = el H 1 ) [n] ) ) just as for F +. Turning to the normal bundle, we study the moving part of the reduced obstruction theory on F + : In families, H E 2 ) equals N + = Ext E 1,F 2 )+Ext E 2,F 1 ) = Ext O X,F 2 ) = H F 2 ) = H O X ) H E 2 ) = C+C H E 2 ). H H 1 L 1 I W ) = C l+2 L 1 H H 1 O W ) L 1. Hence, we find N + = C+C+L 1 H 1 ) [n] L 1 C l+2) [1], where the [1] indicates the torus weight. Similarly, N = C+C+L 1 H 1 ) [n] L 1 C l+2) [ 1],
9 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 9 The normal complexes N ± have virtual rank n l. Taking into account the opposite equivariant weights for the two fixed loci, we write the inverses of the Euler classes of the virtual normal bundles over the two fixed loci as 1 en + ) 1 en ) = 1 ζ) l+2 c + H 1 ) [n] L 1 ), = 1) n l 1+ζ) l+2 c H 1 ) [n] L 1 ). where ζ is the hyperplane class on P. We have used the Chern class notation c + = i 0 c i and c = i 0 1) i c i The calculation. Let ζ also the denote the pull-back of the hyperplane class under the support morphism Q H,χ C 2 ) P, [ 0 E C 2 O X F 0 ] supp F. We calculate by equivariant virtual localization the integrals 0 = I k = 1)n ζ 2k 1 0 2n+2 2k, 1 k n, 2 [Q H,χ C 2 )] red against the reduced virtual class [ QH,χ C 2 ) ] red A2n+1 [QH,χ C 2 ) ] red ). The reduced virtual class admits a canonical lift to the C -equivariant cycle theory. The action of C on ζ is trivial. The C action on 0 is lifted as the equivariant first Chern class of the representation [1]. By examining the overall signs, the two non-vanishing fixed loci F + and F can be seen to have identical contribution. We obtain: I k = ζ 2k 11 ζ)l+2 e H 1 ) [n] L 1) X [n] P c H 1 ) [n] L 1) n = X Pζ 2k 1 1 ζ) l+2 ) c n i H 1 ) [n]) ζ) i s j H 1 ) [n]) [n] 1 ζ) n+j i=0 j 0 = 1) i ζ 2k+i 1 1 ζ) l+2 n j c n i H 1 ) [n]) s j H 1 ) [n]) i,j 0 X [n] P = 1) l n i=0 ) l+2 i 2n l+2 i 2k X [n] c n i H 1 ) [n]) s n+i H 1 ) [n]).
10 10 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE Setting α i = X [n] c n i we have proven the following result. Lemma 3. For all 1 k n, we have n H 1 ) [n]) s n+i H 1 ) [n]) for 0 i n, i=0 l+2 i 2n l+2 i 2k ) α i = 0. In the statement above, we use the follow standard conventions for the binomial: 0 0) = 1, a b) = 0 for 0 a < b, a b) = 0 for b < 0, ) a b = a a 1) a b+1) b! for a > 0, b 0. Since we obtain additionally ch 1 ) [n] ) sh 1 ) [n] ) = 1, α 0 +α α n = Proof of Theorem 1 To prove Theorem 1 for H 1, we must show ) l 2n+2 α n = 2 n. n We already know α n is a degree n polynomial in l, see [23]. The proof will be obtained by combining the following two statements: i) the polynomial α n has roots l = 2n 2, 2n 1,..., 3n 3, ii) the leading term of α n is α n l) = 2n n! ln +... Proof of i). We will use Lemma 3 for the values 6) l = 2n 2,...,3n 3. With the exception of the range 0 l < 2n 2, the other values of l yield trivial relations. Write and relabel l = 2n 2+p, 0 p n 1, β j = α j+p, 1 j n p. We will show that for the values 6) of l we have α n = 0 β n p = 0.
11 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 11 Lemma 3 written for k n k yields the relations n ) p i α i = 0 p i+2k i=0 for all 0 k n 1. When k = 0, the binomials corresponding to i p are equal to 1, while those for i > p are 0. We obtain 7) α α p = 0 = β β n p = 0. When 1 k n 1, the binomials with i p are zero, so the only possible non-zero binomial numbers are obtained for i > p. Writing i = p+j, the above identity becomes n p ) j 8) β j = 0. 2k j j=1 Note that ) ) j 2k 1 = 1) j. 2k j j 1 We collect all the equations 7) and 8) for values k n p 1 derived above. Using the notation of the Lemma 4 below, the system of equations in β j s thus obtained can be written as A n p By Lemma 4, A n p is invertible. Hence, as claimed. β 1 β 2... β n p β n p = 0 = 0. Lemma 4. The n n matrix A n with entries ) 2i 1 a ij = 1) j+1, 0 i,j n 1 j is invertible. Proof. Let A be the n n matrix with entries ) mi a ij =, 0 i,j n 1. j If the m i are distinct, we will prove the invertibility of A via a standard Vandermonde argument. Since A n is obtained up to the signs 1) j+1 which only affect the sign of the determinant) by the specialization the Lemma follows. m i = 2i 1,
12 12 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE If two m s are equal, then the matrix A has two equal rows, so deta vanishes. Therefore, deta is divisible by m i m j ). i<j SinceAhas degree nn 1) 2 in the m s, it suffices to show that the leading term is non-zero. If the leading term were zero, then deta = 0 for all choices of m i. However, for m i = i the matrix A is triangular with 1 s on the diagonal. Proof of ii). For the series A 1 z) in 1), we write A 1 z) = a 11 z +a 12 z 2 +a 13 z 3 +. By evaluating the coefficient of z in 1), we obtain 2la 11 = s 2 H) = 2l = a 11 = 1. The coefficient of z n in 1) yields X N X,H,n = 2la 11) n + lower order terms in l n! = 2n n! ln Statements i) and ii) imply Theorem 1 for primitively polarized K3 surfaces X, H) of degree 2l and Picard rank 1. By 1), Theorem 1 then holds for all pairs X,H) where X is a nonsingular projective K3 surface. 3. Lehn s conjecture for K-trivial surfaces We verify here several statements made in Section 0 concerning Lehn s conjecture: i) Theorem 1 is equivalent to Conjecture 1 for K3 surfaces. ii) Conjecture 1 for K3 sufaces implies equations 3) and 4). iii) Corollaries 1 and 2 hold. Proof of i). Let X,H) be a K3 surface of degree 2l. We must prove N X,H,n z n 1 2w) 2l+6 = 1 6w +6w 2 ) l+2 for N X,H,n = 2 n l 2n+2) n given by Theorem 1 and the change of variables The following Lemma applied to completes the proof of i). z = w1 w)1 2w)4 1 6w +6w 2 ) 3. x = 2z, y = 2w w 2 )
13 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 13 Lemma 5. After the change of variable we have x = y1 2y)2 1 3y) 3, ) l 2n+2 x n = 1 2y)l+3 n 1 3y) l+2. Proof. We carry out a further change of variables 1 2y 1 3y = 1+t = x = t1+t)2. We must show ) l 2n+2 x n = 1+t)l+3. n 1+3t Denote the left side by f l x). Pascal s identity shows the recursion 9) f l+1 x) = f l x)+xf l 2 x). The corresponding recursion for the right side 1+t)l+3 1+3t is easily verified. We prove the Lemma by induction on l. We first verify the base case consisting of three consecutive integers, for instance When l = 4, we must show ) 2n 2 x n = n which may be rewritten as 10) t = l = 4, 3, t)1+3t) = x n+1 n+1 where t = gx) is the solution of the equation 2n 2 n t1+t) 2 = x, ), ) dx 1 = dt dt dx valid in a neighborhood of x = 0 and with g0) = 0. The Taylor expansion 10) for g is equivalent to the value of the n th -derivative d n ) g 2n dxn0) = n 1)!. n 1 The n th -derivative of the inverse function g at 0 is determined by the Schur-Jabotinsky Theorem [8]: g n) 0) equals Since n 1)! coefficient of t 1 in the expansion of x n. x n = t n 1+t) 2n,
14 14 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE the above coefficient has value 2n n 1) as required. Using the identity 1+t 1+3t = 1+t) 3 1+t)1+3t) t1+t)2 = 1+t) 3f 4 x) x and equation 10), we obtain 1+t 1+3t = 1+ = 1+ = f 2 x), x n+1 n+1 2n 2 n x n+1 2n 2 n+1 ) ) 3 ) 2n 2 x n+1 n which verifies the Lemma for l = 2. For n 1, we have the identity ) 2n 1 = 3 ) 2n. n 2 n Taking into account corrections coming from n = 0, we obtain ) 2n 1 x n = 3 ) 2n x n 1 n 2 n 2. Equivalently, f 3 x) = 3 2 f 2x) 1 2 = 3 2 which verifies the remaining base case. 1+t 1+3t 1 2 = 1 1+3t, Proof of ii). We combine equations 1) and 2) for a K3 surface X,H) of degree 2l to obtain Setting l = 0 yields and thus exp2la 1 z)) = exp2la 1 z)+24a 4 z)) = exp24a 4 z)) = 1 2w) 2l+6 1 6w +6w 2 ) l w) 6 1 6w +6w 2 ) 2 1 2w) 2l 1 6w +6w 2 ) l = A 1 z) = 1 2 log Setting as before z = 1 1 4w 2 t1+t)2 +4w2 = 1 6w +6w 2 = 1+t, we obtain equation 3). Equation 4) follows by the same method. 1 4w +4w 2 1 6w +6w 2 ).
15 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 15 Proof of iii). Let A,H) be an abelian or bielliptic surface of degree 2l. Equation 1) and the proof of ii) yield N A,H,n z n = exp2la 1 z)) = Corollary 1 is equivalent to the identity ) z n2n l l 2n 1 = n n 1 1 2w) 2l 1 6w +6w 2 ) l. 1 2w) 2l 1 6w +6w 2 ) l. After the change of variables 2z = x and 2w w 2 ) = y, we obtain ) ) 11) x nl l 2n 1 1 2y l = n n 1 1 3y in the notation of Lemma 5. We have ) dy dx 1 dx = = 1 3y)4 dy 1 2y = d Writing Lemma 5 for l l 5, we obtain ) l 2n 3 x n = 1 2y)l 2 n 1 3y) l 3 and therefore also ) 1 2y = 1 3y)2 1 3y 1 2y dx. ) l 2n 3 x )dx n = 1 2y)l 1 1 2y n 1 3y) l 1d. 1 3y After integration, we obtain x n+1 ) l 2n 3 = 1 1 2y) l n+1 n l 1 3y) l 1 l which yields 11) after the shift n n 1. This proves Corollary 1. Let E,H) be an Enriques surface of degree 2l. Equation 1) yields N E,H,n z n 1 2w) 2l+3 = exp2la 1 z)+12a 4 z)) = 1 6w +6w 2 ) l+1. Using the change of variables 2z = x and 2w w 2 ) = y as above, we obtain N E,H,n 2 n x n = 1 2y)l y) l+1. In particular, by Lemma 5, ) 2 N E,H,n 12) 2 n x n = ) 2l 2n+2 x n = n N X,H,n 2 n x n, where the X, H) is the K3 double covering E with the divisor class determined by pull-back. The proof of Corollary 2 is complete.
16 16 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE Corollary 3. If Y is a minimal elliptic surface and H = mf is a multiple of the fiber class, then Proof. Equation 1) yields n N Y,H,n 2 n x n = ) ) χo Y )/2 2n+2 x n. n 1 2w) N Y,H,n z n 3 = exp12χo Y ) A 4 z)) = 1 6w +6w 2 n ) χoy ). The Corollary follows via the change of variables 2z = x and 2w w 2 ) = y and Lemma Moduli of K3 surfaces 4.1. Tautological classes. Let M 2l be the moduli space of quasi-polarized K3 surfaces X,H) of degree 2l. The Noether-Lefschetz loci define classes in the Chow ring A M 2l,Q). Let NL M 2l ) A M 2l,Q) be the subalgebra generated by Noether-Lefschetz loci of all codimensions). A basic result conjectured in [17] and proven in [1] is the isomorphism NL 1 M 2l ) = A 1 M 2l,Q). Another method of constructing classes in A M 2l,Q) is the following. Let π : X M 2l be the universal surface. Let H X and T π X the universal quasi-polarization canonical up to a twist by an element of PicM 2l )) and the relative tangent bundle. The κ classes are defined by κ a,b = π c 1 H) a c 2 T π ) b) A a+2b 2 M 2l,Q). There is no need to include a κ index for the first Chern class of T π since c 1 T π ) = π λ where λ = c 1 E) is the first Chern class of the Hodge line bundle E M 2l
17 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 17 with fiber H 0 X,K X ) over the moduli point X,H) M 2l. The Hodge class λ is known to be supported on Noether-Lefschetz divisors. 2 The ring generated by λ has been determined in [6]. On the Noether-Lefschetz locus M Λ M 2l corresponding to the Picard lattice Λ, richer κ classes may be defined using all the powers of all the universal line bundles associated to Λ. If H 1,...,H r is a basis of Λ and H 1,...,H r X Λ are the associated universal bundles canonical up to a twist by an element of PicM Λ )) over the universal surface we define the κ Λ classes by X Λ M Λ, κ a1,...,a r,b = π c 1 H 1 ) a1 c 1 H r ) ar c 2 T π ) b) A i a i+2b 2 M Λ ). We define the tautological ring of the moduli space of K3 surfaces, R M 2l ) A M 2l,Q), to be the subring generated by the push-forwards from the Noether-Lefschetz loci of all monomials in the κ classes. By definition, NL M 2l ) R M 2l ) Relations. Consider the universal surface π : X M 2l, and the π-relative Quot scheme Q π dh,χ Cr ) parametrizing quotients C r O X F 0 where F is a rank 0 coherent sheaf on X,H) satisfying c 1 F) = dh and χf) = χ. Just as in Section 1.1, the π-relative Quot scheme admits a reduced virtual class via the π-relative obstruction theory. The push-forwards 13) π γ 0 k [ Q π dh,χ Cr ) ] red ) A M 2l,Q) vanish for all k > 0 and for all choices of cohomology classes γ. There is a natural torus action on Q π dh,χ Cr ) via the torus action on C r. Virtual localization [7] applied to the push-forwards 13) yields relations in R M 2l ). The Noether-Lefschetz loci appear naturally when the curve class dh splits into non-multiples of H. 2 In [17], λ is considered a degenerate Noether-Lefschetz divisor. See [16, Theorem 3.1] to express λ in terms of proper Noether-Lefschetz divisors.
18 18 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE There is no difficulty in writing the resulting relations in R M 2l ) in terms of pushforwards from the π-relative Hilbert schemes of points. To show the relations are nontrivial requires non-vanishing results in the intersection theory of the π-relative Hilbert scheme. A thorougher study of the relations in R M 2l ) obtained by localizing the virtual class of π-relative Quot schemes will appear in [15] Virtual localization. We illustrate the program proposed above with several examples. For simplicity, we will only consider the case d = 1. The relative Quot scheme p : Q π H,χC 2 ) M 2l has relative reduced virtual dimension equal to 2n+1 where n = χ+l. As before, there is a support morphism Q π H,χ C2 ) Pπ H), [ 0 E C 2 O X F 0 ] supp F. Consider the codimension 1 vanishing push-forwards 1) n p 0 2n+2 2k ζ 2k [ Q π 2 H,χC 2 ) ] ) red A 1 M 2l,Q), for 0 k n. We evaluate these expressions via equivariant localization for the torus action induced from the splitting C 2 = C[0]+C[1]. Over the generic point X,H) of the moduli space, the fixed loci over the Quot scheme correspond to kernels of the form E = I Z +I W O X C) where lz) = z, lw) = w, z +w = n and C is a curve in the linear series H. The weights are distributed in two possible ways over the summands. There are additional fixed loci over surfaces in the Noether-Lefschetz divisors corresponding to a nontrivial-splitting [H] = [C 1 ]+[C 2 ] so that E = I Z O X C 1 )+I W O X C 2 ). We will address these later. For now, to describe the generic fixed loci and their contributions, we introduce the following notation:
19 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 19 Over the moduli space M 2l, the bundle V = π H has rank l+2. We write P = PV) and L = O P 1) for the associated relative linear series H over the moduli space and the corresponding tautological line bundle respectively. Over the relative Hilbert scheme X [z] M 2l, we introduce the rank z bundle H [z] = pr 2 pr 1H O Z ), obtained as the push forward from the universal surface of the twisted universal subscheme Z X M X [z] X [z]. The fixed loci are ) A[z,w] = X [z] M X [w] M P. Over the universal surface A[z,w] M X, the universal subsheaf equals E = I Z I W L 1 H 1. The non-reduced obstruction bundle over A[z, w] is found by considering the fixed part of the virtual tangent bundle: Ext E,F) fix = Ext I Z,O Z )+Ext I W L 1 H 1,O X I W L 1 H 1 ). To calculate this bundle explicitly, we follow the method of Section 1.2 for a fixed surface. The only difference is that in the use of relative duality, the relative canonical bundle will yield a copy of the Hodge class K X/M = π E. We only record here the final answer for z 0 and w 0): Obs red = E +E ) O [z] +E L H 1 ) ) [w]. Just as in the case of a fixed surface where almost all fixed loci have a trivial summand, the fixed loci here keep an additional copy of the Hodge bundle. The corresponding contribution ContA[z,w] = mz,w) λ
20 20 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE is a multiple of the Hodge class. In our examples below, mz,w) can be found via a calculation over the Hilbert scheme of a fixed surface. A separate calculation shows that this conclusion also holds for w = 0.) There are however two exceptions, where the formulas above must be modified. These correspond to the two fixed loci F + and F where the splitting of the kernel is E = O X [0] I W O X C)[1] or E = I Z O X C)[0] O X [1] for a curve C in the linear series H. We have The following hold true: Therefore, The obstruction bundle equals so that F + F X [n] M P. Obs red = E L H 1 ) [n] ), [ F + ] vir = [ F ] vir = 1)n e E L 1 H 1 ) [n] ). The virtual normal bundle is N ± = C+E +L 1 H 1 ) [n] L 1 V E ) [±1]. The formulas of Section 1.2 are special cases. 1 en + ) 1 en ) = c L V E) 1 λ = 1) n l c+l V E) 1+λ 1 c + L 1 H 1 ) [n] ), 1 c L 1 H 1 ) [n] ). The localization contributions of the fixed loci F + and F are be equal and take the form q ζ 2k e ) E L 1 H 1 c L V E) 1 ) [n] ) 1 λ c + L 1 H 1, ) [n] ) where q : X [n] M P M 2l is the projection. This expression can be expanded as in Section 1.3. The following push-forwards from the relative Hilbert scheme are central to the calculation: γ i = pr sn+i+1 H 1 ) [n] ) cn i H 1 ) [n] )) A 1 M 2l ), 0 i n. For the sum of contributions of fixed loci dominating M 2l, we obtain n ) ) l+1 2n i 1) l+1 γ i + a k c 1 V) + b k λ, l+1 2k i i=0
21 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 21 for constants a k and b k that depend on l, k, n. For instance, n ) l+1 2n i a k = α i. l+2 2k i i=0 The expression thus obtained must be in the span of the Noether-Lefschetz divisors. Similar relations can written down in higher codimension. The class γ n = pr s2n+1 H 1 ) [n] )) is the codimension one analogue of the Segre integrals which are the subject of Lehn s conjecture. While there are procedures to calculate γ n, we do not yet have a closed form expression. Nonetheless, by the recursions of [5], it can be seen that γ n R M 2l ) is a combination of the classes κ 3,0, κ 1,1, λ with coefficients which are polynomials in l Examples. We present here explicit relations involving κ classes and Noether- Lefschetz divisors. As explained in the introduction of [17], there are two essentially equivalent ways of thinking of the Noether-Lefschetz divisors. Our convention here, which becomes relevant only for the divisor Q below, is that we specify the Picard class. In other words, we consider surfaces for which there exists a class β with given lattice [ ] H 2 β H β H β Degree two. The simplest case is of K3 surfaces of degree H 2 = 2. Relevant to the discussion are the following Noether-Lefschetz divisors: i) The divisor P corresponding to the lattice [ ] ii) The divisor S corresponding to the lattice [ ] The divisor S parametrizes K3s for which the quasi-polarization fails to be ample. By work of O Grady [19], the following isomorphism holds A 1 M 2,Q) = Q [P]+Q [S]. However, there are more tautological classes to consider. In addition to the Hodge class λ, we also have κ 3,0 = π c 1 H) 3 ) and κ 1,1 = π c 1 H) c 2 T π )).
22 22 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE While the classes κ 3,0 and κ 1,1 are canonically defined only up to a choice of a quasipolarization, the difference κ 3,0 4κ 1,1 is independent of choices. There are two relations connecting the four tautological classes Proposition 1. We have κ 3,0 4κ 1,1, λ, [P], [S]. κ 1,1 4κ 3,0 18λ+12[P] = 0, κ 1,1 4κ 3, λ 24 5 [P] 3 [S] = Proof. The relations are obtained by considering the following numerics: i) For the first relation, we use the relative Quot scheme of short exact sequences where 0 E C 2 O X F 0 c 1 F) = H, χf) = 0. The relative virtual dimension equals 3, and we evaluate the pushforward p 0 4 [Q π H,0 C2 )] red ) A 1 M). ii) For the second relation, we use the numerics c 1 F) = H, χf) = 1. The relative virtual dimension equals 5, and we consider the pushforward p 0 6 [Q π H,1 C2 )] red ) A 1 M). We provide details only for case i) since ii) is parallel. Over a generic surface X,H), the total contribution can befound by the method of Section 4.3. The answer is 2γ 1 4c 1 V) = 2κ 3,0 4c 1 V) = 4κ 3,0 κ 1,1 +6λ. 3 Grothendieck-Riemann-Roch was used to express c 1 V) = 3λ 2 + κ 1, κ 3,0 6. Over the locus P, the classes β, H β are effective with The torus fixed kernels are β H β) = 1, β 2 = 0, H β) 2 = 0. E = O C 1 )+O C 2 )
23 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 23 for curves C 1 and C 2 in the linear series β and H β. There are two fixed loci, with identical contributions, obtained by switching the weights. The multiplicity of the divisor P in the final calculation equals 2 over each fixed locus. The 2 can be confirmed by working over a fixed surface. Each fixed locus is isomorphic to H 1 H 2 = P 1 P 1. The fixed part of the tangent obstruction theory equals Ext O C 1 ),O C1 )+Ext O C 2 ),O C2 ). The Hodge factor of the reduced theory cancels the additional Hodge class coming from the normal bundleof the divisor P in the moduli stack, see [19]. The virtual class of the fixed locus agrees with the usual fundamental class. Similarly, the normal bundle is found by considering the moving part where N = Ext E,F) mov = N 1 [1]+N 2 [ 1] N 1 = Ext O C 1 ),O C2 ) = H OC 1 ) C2 ) = H OC 1 )) H OC 1 C 2 )) N 2 = Ext O C 2 ),O C1 ) = H OC 2 ) C1 ) = H OC 2 )) H OC 2 C 1 )). In families, these formulas also include the the tautological bundles L 1 and L 2 over the linear series. We have N 1 = H H 1 ) L 1 H H 1 H2 1 ) L 1 L 1 2 = L 1 C 2 L 1 L 1 2 and analogously N 2 = L 2 C 2 L 2 L 1 1. The final multiplicity becomes 1 P 1 P 1 en) = 1+ζ 1 ζ 2 ) 1 ζ 1 +ζ 2 ) P 1 P 1 1+ζ 1 ) 2 1+ζ 2 ) 2 = Degree four. Our next example concerns the case of quartic K3 surfaces with H 2 = 4. We single out three Noether-Lefschetz divisors: i) The divisor P corresponding to the lattice [ ] ii) The divisor Q corresponding to the lattice [ ]
24 24 A. MARIAN, D. OPREA, AND R. PANDHARIPANDE iii) Finally, there is a the divisor S corresponding to the lattice [ ] Here, the linear series H fails to be ample, and the associated birational map contracts a 2) curve to an ordinary double point. A K3 surface in the complement of the union of the three divisors P, Q, S is necessarily a nonsingular quartic in P 3. Again by work of [19], we have an isomorphism The class A 1 M 2,Q) = Q [P]+Q [Q]+Q [S]. κ 1,1 2κ 3,0 is independent of the choice of polarization. The following relations are obtained: Proposition 2. We have 10κ 1,1 2κ 3,0 ) 38λ+8[P] 2[Q] 3[S] = 0, 4κ 1,1 2κ 3,0 ) 26λ+16[P]+2[Q] [S] = 0. The proof of Proposition 2 uses the relative Quot scheme of short exact sequences where the numerics are chosen so that 0 E C 2 O X F 0 c 1 F) = H, χf) = 0. The relative virtual dimension equals 5. The first relation is obtained by evaluating the pushforward p 0 6 [Q π H,0C 2 )] red ) A 1 M), while second relation is obtained by evaluating p 0 4 ζ 2 [Q π H,0 C2 )] red ) A 1 M). The details are similar to the proof of Proposition Conjecture. The main conjecture suggested by the abundance of relations obtained by localizing the virtual class of π-relative Quot schemes is the following. Conjecture 2. For all l 1, we have NL M 2l ) = R M 2l ). If true, Conjecture 2 would lead to a much simpler picture of the additive structure of R M 2l ) since good approaches to the span of the Noether-Lefschetz classes are available [2, 10]. We further speculate that the relations obtained by localizing the virtual class of π-relative Quot schemes are sufficient to prove Conjecture 2.
25 SEGRE CLASSES AND HILBERT SCHEMES OF POINTS 25 References [1] N. Bergeron, Z. Li, J. Millson, and C. Moeglin, The Noether-Lefschetz conjecture and generalizations, arxiv: [2] R. E. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J ), [3] E. Cotterill, Geometry of curves with exceptional secant planes: linear series along the general curve, Math. Zeit ), [4] M. Dürr, A. Kabanov and C. Okonek, Poincare invariants, Topology ), [5] G. Ellingsrud, L. Göttsche, M. Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom ), [6] G. van der Geer and T. Katsura, Note on tautological classes of moduli of K3 surfaces, Mosc. Math. J ), [7] T. Graber, R. Pandharipande, Localization of virtual classes, Invent. Math ), [8] E. Jabotinsky, Representations of functions by matrices, Proc. Amer. Math. Soc ), [9] F. Janda, Tautological relations in moduli spaces of weighted pointed curves, arxiv: [10] S. Kudla, J. Millson, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Pub. IHES ), [11] P. Le Barz, Formules pour les multisecants des surfaces algebriques, L Ens. Math ), [12] M. Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math ), [13] A. Marian and D. Oprea, Virtual intersections on the Quot scheme and Vafa-Intriligator formulas, Duke Mathematical Journal ), [14] A. Marian, D. Oprea, and R. Pandharipande, The moduli space of stable quotients, Geometry and Topology ), [15] A. Marian, D. Oprea, and R. Pandharipande, in preparation. [16] D. Maulik, Supersingular K3 surfaces for large primes, Duke Math. J ), [17] D. Maulik and R. Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory, A celebration of algebraic geometry, , Clay Math. Proc., 18, Amer. Math. Soc., Providence, RI, [18] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. Math ), [19] K. O Grady, Moduli of abelian and K3 surfaces, Ph.D. Thesis, Brown University, [20] R. Pandharipande, The kappa ring of the moduli space of curves of compact type, Acta Math ), [21] A. Pixton and R. Pandharipande, Relations in the tautological ring of the moduli space of curves, arxiv: [22] D. Schultheis, Virtual invariants of Quot schemes over del Pezzo surfaces, Ph.D. Thesis, UC San Diego 2011). [23] A. S. Tikhomirov, Standard bundles on a Hilbert scheme of points on a surface, Algebraic geometry and its applications, Yaroslavl 1992, Aspects of Mathematics, Vol. E25, Vieweg Verlag, [24] A. S. Tikhomirov and T. L. Troshina, Top Segre class of a standard vector bundle E 4 D on the Hilbert scheme Hilb 4 S) of a surface S, Algebraic geometry and its applications, Yaroslavl 1992, Aspects of Mathematics, Vol. E25, Vieweg Verlag, [25] A. N. Tyurin, The Spin-polynomial invariants of the smooth structures of algebraic surfaces, Iz. An SSSR ), [26] Z. Wang, Tautological integrals on Hilbert schemes of points on curves, arxiv: Department of Mathematics, Northeastern University address: a.marian@neu.edu Department of Mathematics, University of California, San Diego address: doprea@math.ucsd.edu Department of Mathematics, ETH Zürich address: rahul@math.ethz.ch
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