Higher rank Segre integrals over the Hilbert scheme of points

Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S. When s=1, the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of tautological bundles for all ranks s over all K-trivial surfaces. Furthermore, in rank s=2, the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler characteristics over the Hilbert schemes of points.

Tautological integrals over the Hilbert scheme of points or over their geometric subsets emerge often in enumerative geometry. We mention three situations studied previously: (i) the count of n-nodal curves in fixed linear system P n ⊂ |L| where L → S is sufficiently positive. These counts were recast by Göttsche [G1] in terms of Hilbert schemes. Specifically, for a suitable geometric subscheme W n ֒→ S [3n] birational to S [n] , the Severi degrees are encoded by the series z n Wn c (L [n] ).
(ii) the Verlinde series of holomorphic Euler characteristics [EGL]: Here the line bundle () (n) is pulled back from the symmetric product, and E denotes − 1 2 of the exceptional divisor on the Hilbert scheme.
1 all types of tautological integrals considered over the past few decades, closed formulas for the remaining series U 3 , U 4 have proven very difficult to calculate or even conjecture. Remarkably, the Segre geometry (iii) is the only nontrivial case known so far; the details are explained in Section 1.3.
1.2. Higher rank. We undertake the study of the Segre series in higher rank. To set the stage, let V → S be a vector bundle of rank s, inducing a tautological rank sn vector bundle over the Hilbert scheme via the assignment The construction extends to K-theory. For each K-theory class α ∈ K(S), there is an associated K-theory class defined via locally free resolutions. We are interested in the Chern numbers of α [n] , but it will be more convenient to work with Segre classes. Fixing α ∈ K(S), we consider its associated Segre series S α (z) = ∞ n=0 z n S [n] s(α [n] ) (1) The above factorization follows by [EGL]. The five series are independent of the surface S, and depend on α only through the rank s.
1.3. Lehn's conjecture. Let us assume first that rank α = 1 and α is represented by a line bundle Then, the series A 0 is absent from (1). Lehn [L] conjectured closed formulas for all four series A 1 , A 2 , A 3 , A 4 . In the rephrasing of [MOP1], after the change of variables the universal Segre series are given by A 2 (z) = (1 + 2t) 3 2 · (1 + 6t) − 1 2 , A 3 (z) = 1 2 · (1 + 2t) −1 · √ 1 + 2t + √ 1 + 6t , A 4 (z) = 4 · (1 + 2t) 1 2 · (1 + 6t) The series A 1 and A 2 were confirmed via the study of the virtual geometry of the Quot scheme of K-trivial surfaces in [MOP1]. Using Reider type techniques, Voisin [V] proved the vanishing of certain Segre integrals in the case of the blowup of a K3 surface in one point, and showed these vanishings determined uniquely the series A 3 , A 4 . The series were shown to have Lehn's conjectured expressions via the residue calculations of [MOP2]. As a result, the Segre series are calculated for all line bundles over nonsingular projective surfaces S.
Remark 3. Even in rank 1, Theorem 1 gives additional information not covered by Lehn's conjecture. While the series A 1 , A 2 were predicted by Lehn's formula, was absent from the line-bundle setup of the conjecture.
Remark 4. For nonsingular projective curves, the generating series of higher rank Segre integrals was determined in [MOP2]. A change of variables similar to (3) was needed to express the answer in closed form.
Remark 5. The expressions A 1 , A 2 determined by Theorem 1 were subsequently connected by Mellit with certain generating series of Hurwitz numbers in genus zero [M].
1.5. The series A 3 , A 4 . For surfaces S which are not K-trivial, and an arbitrary class α ∈ K(S), the universal series A 3 , A 4 appear in the Segre generating function S α (z) and are very difficult to evaluate.
We calculate the two remaining universal series for rank α = 2, thus proving a full rank-two analogue of Lehn's formulas. Thus, we provide a second nontrivial situation for which all series involved are determined. Using the change of variables

Theorem 1 gives
To describe the remaining two series, let y(t) be the unique real solution of the quartic equation We have Theorem 2. For rank α = 2 the remaining universal Segre series are Based on these expressions, we also formulate Conjecture 1. For all ranks, the functions A 3 , A 4 are algebraic.
Remark 6. Finding the series A 3 and A 4 for α of arbitrary rank remains an open question. In addition to Lehn's formula for rank one, and the substantial rank two analysis carried out here, a few cases are trivially known: • the case rank α = −1 is immediate. Set α = −L for some line bundle L → S.
• for rank α = 0, we have A 4 (z) = 1 just by setting α = 0. Regarding the remaining series, we state the following Conjecture 2. In rank 0, we have 1.6. The Verlinde generating function. We now turn to the generating series of holomorphic Euler characteristics of tautological line bundles over the Hilbert scheme. We set where the line bundle () (n) is pulled back from the symmetric product, and E denotes − 1 2 of the exceptional divisor on the Hilbert scheme. Since V α only depends on det α, only four power series are needed in the factorization The different form of the exponents used here relative to (1) is justified by the fact that with the current choices, we have the following symmetries This was noted in Theorem 5.3 in [EGL] as a consequence of Serre duality. Also by [EGL], we explicitly know B 1 and B 2 . For The series B 3 , B 4 remain mysterious in general. The cases r = 0 and r = ±1 are immediate exceptions, obtained in [EGL], Lemma 5.1. Indeed for r = 0, we have For all surfaces, motivated by strange duality, Johnson [J] predicted a connection between the Segre and Verlinde series under an undetermined change of variables z ↔ w, and the shift (6) r = rank α − 1.
These predictions were made precise in [MOP2] where the unknown change of variables was proposed: Using Theorem 1 we show under suitable numerics an equality between Verlinde numbers and tautological Chern integrals on Enriques surfaces, see Proposition 4. This is consistent with strange duality.
Since the Segre series is now calculated for rank α = 1 (Lehn's formula) and rank α = 2 (in Theorem 2), this precise conjectural Verlinde-Segre relationship gives further predictions for the unknown Verlinde universal series B 3 and B 4 .
For r = ±2, the formulas for B 3 and B 4 are captured by Conjecture 2 of [MOP2]. Indeed for r = 2, setting 1 Thanks to Theorem 2, we obtain the following predictions about the Verlinde series for r = ±3. Let Y be the unique real solution of the quartic equation We have Conjecture 3. For the Verlinde series V α with r = 3, setting The expressions for r = −3 are obtained via the symmetry (5).
Finding the general expression for the unknown series B 3 and B 4 for arbitrary r, thus determining all rank 1 Verlinde numbers, is a central question in the enumerative theory of Hilbert schemes of points on surfaces.
Remark 7. Due to the rank shift (6), the Serre duality symmetry s → −s on the Verlinde side translates into a conjectural transformation rule for the remaining unknown Segre universal series A 3 , A 4 , as s → −s − 2.
In particular, we can also state predictions for the Segre series A 3 , A 4 in rank α = −3 and rank α = −4 from the series in rank α = 1 and rank α = 2. The exact expressions are more cumbersome, but they support Conjecture 1. 1.7. Strategy of proofs. We first explain how Theorem 1 is derived. To find the three universal series A 0 , A 1 , A 2 , we pick α to be the class of a suitable vector bundle V → S over a K3 surface S. As witnessed by (4), the Segre integrals are generally very complicated. It is a key observation that the answers take a simpler form for sheaves with small deformation spaces. The most beautiful formulas are obtained for spherical vector bundles V and for vector bundles with isotropic Mukai vectors. Let Set χ = χ(S, V ). We show: Theorem 3. Let S be a K3 surface, and let V → S be a rank s = r − 1 vector bundle.
Theorem 3 is proven by a direct geometric argument using Reider techniques [R], also employed in rank 1 in [V]. The crucial insight here is the identification of the optimal geometric setup for which the complicated Segre integrals become manageable. We will then show by a residue calculation that Theorems 1 and 3 are equivalent.
The analysis is quite intricate for Theorem 2, requiring in particular delicate excess calculations for Segre classes. The key statements are captured by the following Theorem 4. Let π : S → X be the blowup of a K3 surface X at a point, with exceptional divisor E. Let V 0 → X be a rank 2 bundle whose Mukai vector satisfies v 0 , v 0 = −2.
We finally show by a residue calculation that Theorems 2 and 4 are equivalent.
When π is the universal family of the moduli of polarized surfaces π : S → M, the κ-classes thus defined generate the tautological ring R ⋆ (M) analogous to the wellstudied tautological ring R ⋆ (M g ) of the moduli of curves. Finding relations between the κ-classes in R ⋆ (M) is a very interesting problem.
In the case of the moduli of K3 surfaces, a strategy for κ-relations was laid out in [MOP1] via the study of the virtual class of the Quot scheme; a different approach via Gromov-Witten theory was pursued in [PY]. The discussion however makes sense for arbitrary polarized surfaces as well. In this approach, the center stage is taken by the calculation of the push-forwards in terms of the classes κ[a 1 , . . . , a ℓ , b]. In the K-trivial case, or more ambitiously for arbitrary surfaces, it becomes important to obtain explicit formulas, thus generalizing the results of this paper. 2. K3 surfaces 2.1. Residue calculations. Our first goal is to prove Theorem 1. We begin by explaining how the special formulas of Theorem 3 are predicted by the series in Theorem 1. Conversely, we will prove in Section 2.4 that these predictions are equivalent to the statement of Theorem 1. We keep the same notation as in the introduction. Let S be a K3 surface. For a vector bundle as in Section 1.7. Recall that s = rk V and r = s + 1. Taking the K-theory class α to be V , the statement of Theorem 1 becomes For convenience, we define We then need to prove An important observation is that the quantity (7) is half the dimension of the moduli space M v of stable sheaves of type v on the K3 surface S, We can express the individual Segre integrals as the residues Substituting equation (8) in the above residue and using the change of variables (3) we are equivalently seeking to prove the following formula for the top Segre classes: The formula yields a remarkably simple answer in the following two cases, leading to the statement of Theorem 3.

2.2.
Vanishing results. We now prove the formulas above, thus establishing Theorem 3. The proof is guided by the simple form of the expected answers. Throughout this Section, we let S be a K3 surface of Picard rank one, We assume that v, v = −2 or v, v = 0.
2 H-stability is Gieseker stability with respect to the polarization H.
In both of these cases, we know the moduli space of H-stable sheaves with Mukai vector v is nonempty, see [Y3], Theorem 0.1 for a general result. The moduli space is either a point, or a K3 surface, see Theorems 1.4 and 3.6 in [Mu]. Since s > 1, the locus of locally free sheaves V in the moduli space with Mukai vector v is nonempty. Nonemptiness is obtained by invoking Yoshioka's classification of Mukai vectors yielding moduli consisting entirely of nonlocally free sheaves in Proposition 0.5 of [Y3]. His classification does not include vectors v as above.
Under these assumptions, we show the following vanishings.
Proposition 1. We have . This is equivalent to the surjectivity of the natural vector bundle map We will show that this is the case under our numerics in Proposition 2 below. By the definition of Segre classes, whenever the evaluation map is surjective, we have the vanishing

In our situation
Indeed, by Serre duality and stability, The vanishing (10) is established in the proof of Proposition 2 below, where it is argued that there do not exist nontrivial extensions Therefore, (n − 1)-very ampleness of V implies the vanishing In particular, we obtain Proposition 2. We have Proof. The evaluation map (9) The cohomology group (11) is Serre dual to Ext 1 (I Z , V ∨ ) which is the space of extensions By assumption, we have a non-split extension.
1 of [Y2] shows that the middle term E of any non-split extension For the benefit of the reader, let us recall the argument in [Y2] in our context. Since Assuming now that E is not H-stable, let G ֒→ E be the maximal semistable destabilizing subsheaf. We have We see that G cannot be a subsheaf of the kernel V ∨ since this would contradict the H-stability of the latter. Therefore, we have a nonzero morphism Therefore, we have a = 0 and c 1 (G) = 0 .
Furthermore, if nonzero, the kernel K of φ is a subsheaf of V ∨ of slope greater than or equal to zero, contradicting the H-stability of V ∨ . Thus, we have and the associated sequence of extension groups The first group is zero by Serre duality since Q is supported at finitely many points. We conclude that f is injective. However, the image of the extension (12) In both cases of Proposition 2, we obtain Therefore, by Serre duality, Since E is H-stable and therefore simple, we have a contradiction. Thus, (11) does not hold. The proofs of Proposition 2 and Proposition 1 are therefore complete.
2.3. Proof of Theorem 3. We consider case (i) of the Theorem. Fix the rank s > 1 throughout. 3 By Theorem 4.1 in [EGL], over K3 surfaces, the Segre integral is given by a universal polynomial of degree n in c 1 (V ) 2 and c 2 (V ). When v, v = −2 both c 1 (V ) 2 and c 2 (V ) can be expressed in terms of χ = χ(V ), and therefore for a degree n universal polynomial P n .
For each χ > s, let (S, H) be a K3 surface of Picard rank one Pic(S) = ZH and of genus Let V be the unique H-stable vector bundle with Mukai vector v = (s, H, χ − s) whose existence we noted in the beginning of Section 2.2. Since v, v = −2, Proposition 1 (i) shows that the top Segre class of V [n] vanishes when rn ≤ χ < (r + 1)n.
These n values of χ are the n roots of the polynomial P n . Therefore, s 2n (V [n] ) = c · χ − rn n for some constant c. We will identify the constant c = r n by proving The argument is most naturally expressed by rewriting (2) in exponential form,

Restricting to the line
for power series U and T . Let Extracting the coefficient of z n in the above expression yields In particular, for n = 1, we obtain Direct calculation shows s 2 (V ) = rχ − r 2 so that u 1 = r. After substitution, we find the leading term to be u n 1 n! = r n n! .
For part (ii) of the Theorem, Proposition 1 (ii) gives only n−1 roots of the polynomial expressing the Segre integral. Combined with the leading term calculations, we conclude that if v, v = 0 we have for some constant c. We will prove c = n −r + 1 r by computing the next term in the Segre polynomial in χ. Indeed, for the right hand side, the χ n−1 coefficient is easily seen to be We compute the χ n−1 coefficient on the left. As before, we have
When n = 2, for isotropic vectors v, v = 0, equation (4) simplifies to Using the above asymptotics, we have Thus, for arbitrary n, the χ n−1 -coefficient equals Comparison of (13) and (14) yields the requisite value for the constant c, completing the argument.
2.4. Proof of Theorem 1. By [EGL], the generating series of Segre integrals takes the form for three power series a 0 (z), a 1 (z), a 2 (z) ∈ Q [[z]]. We show that these power series are as claimed by Theorem 1: Fix an integer χ > s, and let g = s(χ − s). Consider (X, H) a K3 surface of genus g, and let V be a vector bundle with Mukai vector We calculate The residue calculations of Section 2.1 and the first part of Theorem 3 together imply Since χ is arbitrary, we obtain We now derive an additional equation using isotropic bundles corresponding to the second part of Theorem 3. To this end, let g = s(χ − s) + 1. Let We then calculate Repeating the above argument for the new numerics, we replicate equation (15)  Constraints (15), (16), (17) show that A i = a i for 1 ≤ i ≤ 3, as claimed.
2.5. Chern classes. Let α → S be a K-theory class of rank s on a K3 surface S, and let s = r + 1. As a corollary of Theorem 1, we also obtain expressions for the Chern classes of tautological bundles, After the change of variables the series are given bỹ Remark 8. The rank s = 2 case (corresponding to r = 1) is easily calculated from geomety by assuming the existence of a transverse section of V , see [J]. The derivation using our formulas is also simple, We find which is the correct answer.
The evaluation ofÃ 0 ,Ã 1 , andÃ 2 for Chern classes follows by regarding the Segre integrals of Theorem 1 as functions on the K-theory of the surface S which depend polynomially on rank α = s, c 1 (α) 2 , c 2 (α) , see [EGL]. Having established these polynomials for positive s > 0, we may then also allow s to be negative, and replace α by −α.

Abelian and Enriques surfaces
3.1. Abelian surfaces. Having established the K3 case, we can also determine the Segre integrals over abelian or bielliptic surfaces S. Indeed, by the universality results of [EGL], we have Residue calculations give the coefficients of the right hand side where v, v = 2d. In particular, we obtain the following generalization of the s = 1 result of [MOP1].
Proposition 3. Let V → S be a vector bundle of rank s = r − 1 on an abelian or bielliptic surface S. If v, v = 0, then

Enriques surfaces and strange duality.
When S is an Enriques surface, the link between the Segre and Verlinde series yields individual equalities of intersection numbers, as the numerical data can be suitably matched. The Proposition below parallels Conjecture 2.2 of [J] formulated for del Pezzo surfaces. The corresponding result does not hold for other K-trivial surfaces.
Proposition 4. Let V → S be a vector bundle of rank s = r + 1, determinant L, and χ(V ) = (r − 1)n + 1. We have The left hand side interprets enumeratively the Verlinde number on the right as an integral over the Hilbert scheme -counting the expected (finite) number of quotients V ∨ → I Z with Z a subscheme of length n. Implications of Proposition 4 for strange duality over K3 and Enriques surfaces will be taken up elsewhere.

K3 blowups
In order to determine the functions A 3 , A 4 , we need to consider surfaces which are not K-trivial. We will look at two different families of examples over the blowup of a K3 at a point. These examples are stated in Theorem 4. We will prove this theorem first. As before, Reider-type arguments play a key role in the calculation. In addition, several excess intersection calculations are needed. Afterwards we show Theorem 2: the integrals calculated in Theorem 4 give the rank 2 series for all surfaces.

Stability in extensions. Let
π : S → X be the blowup of a K3 surface X at one point. We assume that Pic X = ZH. Then where E is the exceptional divisor on S and H denotes the pullback to S of the ample Picard generator on X. Note that H is a nef line bundle on S.
The notion of H-stability and H-semistability have the usual meaning: Definition 1. We say that a torsion-free sheaf V on S is H-stable if for any nonzero subsheaf G of strictly smaller rank, we have For the notion of H-semistability the inequality is not required to be strict.
We first prove several statements regarding vanishing of cohomology which will be useful in the proof of Theorem 4. We consider a vector bundle V 0 → X with Mukai vector v 0 such that The existence of the bundle V 0 was noted in the beginning of Section 2.2: V 0 is the unique point in the moduli space of Gieseker H-stable bundles with Mukai vector v 0 . Such vector bundles are necessarily rigid, that is In order to compute Segre classes of V [n] → S [n] , we analyze the surjectivity of the evaluation map: Thus, for a zero dimensional subscheme Z of length n, we investigate the vanishing of Assume that a non-zero element exists in the extension group. We prove Lemma 1.
(i) Assume k ≥ n − 1. For any nonsplit extension The same conclusion holds, unless the scheme Z is supported on the exceptional divisor E.
Proof. Assume G ֒→ W is an H-semistable destabilizing subsheaf. We then have As a consequence, we see that G cannot be a subsheaf of the kernel V ∨ ⊗ E since this would contradict the H-stability of the latter.
Thus we have a nonzero morphism φ : from which we conclude that a = 0 and µ G = 0. Furthermore, if nonzero, the kernel K of φ is a subsheaf of V ∨ ⊗ I Z of slope greater than or equal to zero, contradicting the H-semistability of V ∨ ⊗ E. Thus we have in fact that φ : G → I Z is injective, so for a zero-dimensional subscheme U ⊂ S and m ≥ 0. In particular Z is supported on the exceptional curve E and on U . We turn to the exact sequence 0 → G → I Z → Q → 0 and the associated sequence of extension groups The image of the extension (24) under α is trivial since the resulting extension is seen to be split. Turning to the first extension group, we have Recall the notation V = π ⋆ V 0 ⊗ E −k . We claim that This would imply α is injective, and in turn that the original extension e splits -a contradiction.
Note that Q is supported on the exceptional divisor and a finite number of points in S. To prove the claimed vanishing, let us first assume that U is empty. The argument is best illustrated by the case m = 1. In this case, the defining sequence shows that Q = I Z/E is the ideal sheaf of n points on the exceptional divisor. Since we see that We now consider the case of arbitrary m, continuing to assume that U = ∅. We let W ℓ = ℓE be the scheme defined by the ideal sheaf E −ℓ . Set Z m = Z and Q m = Q. The defining exact sequence Inductively define the scheme-theoretic intersection for 1 ≤ ℓ ≤ m − 1, and let Q ℓ be given by the exact sequence We will show inductively that for all ℓ ≤ m. The base case ℓ = 1 was verified above. For the inductive step, form the diagram Note that the support of M ℓ has length at most n. As we already noted If ℓ ≥ 1 and k + 1 ≥ n, we have just enough positivity to ensure that the morphism is surjective. Using the first row we conclude Using the first column and invoking the inductive hypothesis, we conclude This completes the argument when U is empty. For the general case, let T denote the scheme with ideal E −m ⊗ I U , and recall W m had ideal E −m . The defining exact sequence gives By composing the first map with the canonical restriction O T → O Wm we obtain an exact sequence is supported on at most n points. Then the previous argument applied to the exact sequence (27) gives the vanishing H 1 (V ⊗ Q) = 0. This finishes the proof when k = n − 1. When k = n − 2, the same argument carries through, unless m = 1 and Z is contained in E, as one can easily check going through the details, in particular by examining (26).
Lemma 2. If W is H-stable, then for any k ≥ 0, the dimension of Hom(W, W ⊗ E k ) equals 1.
Proof. The H-stability of W implies that any nonzero homomorphism φ : W → W ⊗ E k is injective. This can be seen as usual by examining the kernel and image of φ. Let λ be an eigenvalue of φ at a point p in S where W is locally free and p ∈ E. Let I be the canonical homomorphism W → W ⊗ E k obtained by W -twisting the unique section O → E k . We claim φ = λI.
By the first line of the proof, the morphism ψ : W → W ⊗E k must be injective. Consider the induced morphism det ψ : det W → det W ⊗ E kr .
Writing as before I for the det W -twisting of the unique section of E kr , we conclude det ψ = µI for some constant µ. By construction, det ψ vanishes at p. However I only vanishes along E and p ∈ E. This shows that µ = 0, so det ψ = 0. This contradicts the fact that ψ is injective.
Proof. We continue working with the exact sequence (24). If Z ⊂ E, W must be H-stable by Lemma 1. Using Lemma 2, we compute On the other hand, from the defining exact sequence (24), we calculate: Note now that if k = n − 2, then which is a contradiction. This establishes (i). If k = n − 1, then χ(W, W ) = 2. Since Ext 0 (W, W ) = Ext 2 (W, W ) = C we find that Ext 1 (W, W ) = 0 so W is rigid. This establishes (ii).
Note that the same argument for Z = ∅ shows that H 1 (V ) = 0, while H 2 (V ) = 0 follows by stability.
We will analyze the two situations (i) and (ii) in Propositions 5, 6, and 7 below. 4.2. Excess calculations for k = n−2. The goal of this section is to prove the following result which corresponds to Theorem 4 (i).
Proposition 5. Let S be the blowup of a K3 surface at one point. Let with V 0 a rank 2 bundle satisfying (21). Assume furthermore that χ(V ) = 4n − 1. Then Proof. We compute s 2n (V [n] ) as an excess intersection over the nonsurjectivity locus of the evaluation map (23). By Lemma 3, this nonsurjectivity locus consists of those Z with Z ⊂ E, or in other words For the excess calculation, we need to interpret the Segre class s 2n in connection with the top Chern class of a vector bundle, where the excess formula is easier to understand. This connection is as follows, cf. Fulton [F].
By [F], Proposition 14.2.2, Here E and V [n] are pulled back to the product from G(2n, 4n − 1) and S [n] respectively. The vector bundle E ∨ ⊗ V [n] has a natural section s. This is induced by the morphism The zero locus D of the section s is expected to be zero-dimensional, but let us suppose it is in excess with dimension d, so that D ⊂ S [n] × G(2n, 4n − 1), with normal bundle N.
The excess intersection formula of Section 14.4 in [F] reads where π : D → S [n] . We now compute the right hand side of equation (28). By definition, We describe D in concrete terms. Observe that and set The argument of Lemma 3 shows that H 1 (V ⊗ O(−E)) = 0. Therefore Observe next that is surjective. The kernel is of dimension (4n − 1) − 2n = 2n − 1, hence it cannot contain a subspace W of dimension 2n. Furthermore, for Z ⊂ E, the restriction is injective. Therefore, Consequently, we have This identification holds scheme-theoretically. Indeed, it is easy to check that the above pointwise arguments can also be carried out in families. The key observation is that cohomology and base change commute for all relative constructions involved.
In K-theory, the normal bundle to D is Here 2n − 2 is the dimension of the quotient H 0 (V )/Λ 0 and the two summands are restricted to D. We now calculate Let ζ denote the hyperplane class on E [n] ≃ P n and h denote the hyperplane in the Grassmannian G(2n, Λ 0 ) ≃ P 2n . We have The last two formulas for the total Chern classes of tautological vector bundles over the Hilbert scheme of points on P 1 are explained for instance in the proof of Theorem 2 of [MOP2]. Completing the calculation, 4.3. The case k = n − 1. Consider now the case k = n − 1, so that (21), and χ(V ) = 4n − 1. By Lemma 3 (ii), all nontrivial extensions must have the middle term W a rigid H-stable sheaf of rank 3. Of course, We record the following Lemma 4. There are no rigid H-stable sheaves of rank 3 satisfying (30) when n ≡ 0 mod 3.
When n ≡ 0 mod 3, the sheaf W must necessarily be of the form where W 0 → X is a bundle over the K3 surface with (31) rank W 0 = 3, c 1 (W 0 ) = −H, W 0 rigid and Gieseker H-stable.
Proof. As noted in [G2], [Y1], stability with respect to the non-ample divisor H on the blowup is equivalent to H − ǫE-stability for ǫ sufficiently small. Thus the moduli space of H-stable sheaves on the blowup admits a natural scheme structure. In general, the moduli space M S H (3, −H + 2nE, c 2 ) of H-stable sheaves on the blowup has expected dimension 6d − 16 where d denotes the discriminant. For rigid moduli spaces, we have d = 8 3 . The corresponding moduli of H-stable sheaves is zero dimensional and consists of isolated points. To count the stable rigid sheaves, we compute the Euler characteristic of the moduli space.
To this end, we use the blowup formulas of Proposition 3.4 in [Y1] or Proposition 3.1 (2) in [G2]. The generating series of Euler characteristics on the blowup is computed in terms of the same series on the underlying K3 surface X as where x, y ∈ Z + 2n 3 . To complete the proof, we compute the constant term in the above expression. If n ≡ 0 mod 3, there is no constant term due to the factor q x 2 +xy+y 2 .
For n ≡ 0 mod 3, the constant term is 1. The moduli space consists of a single rigid sheaf. Note that when χ(V ) = 4n − 1, a direct calculation shows that there exists a rigid sheaf W 0 on X such that W = π ⋆ W 0 ⊗ E 2n 3 on S has the numerics determined by the exact sequence (29). Specifically the numerical assumptions on V require H 2 = 4n 2 + 12n − 14 and we select W 0 to be in the moduli space with Chern numbers This W is then the only sheaf in its moduli space.
By the Lemma, if n ≡ 0 mod 3 the extension W cannot exist. Consequently, we have for all Z. This shows that the evaluation map (23) over the Hilbert scheme is surjective, and we have obtained the following result corresponding to Theorem 4 (ii): with χ(V ) = 4n − 1 and V 0 a rank 2 bundle as in (21), then s 2n (V [n] ) = 0 for n ≡ 0 mod 3.
We now focus on the remaining statement of Theorem 4. We show with χ(V ) = 4n − 1 and V 0 a rank 2 bundle as in (21), then s 2n (V [n] ) = 1 for n ≡ 0 mod 3.
Proof. As usual, the top Segre class is supported on the locus D where the evaluation map (23) is not surjective. We now identify D as a subvariety of the Hilbert scheme S [n] . The nonsurjectivity locus consists of subschemes Z such that in other words corresponding to the existence of nontrivial extensions When n = 3ℓ for ℓ ∈ Z, as seen in Lemma 4, all such extensions have the same middle term, the unique stable rigid rank 3 vector bundle W on S with numerics specified by the lemma. A direct calculation establishes further that We now show that To start, note the useful vanishing which will be established in Lemma 5 using Reider arguments. By Serre duality and stability From the exact sequence (32), upon tensoring with V , these vanishings imply By (34), for any given Z in the degeneracy locus, the basic extension (32) is unique. Conversely, any non-zero morphism must be injective by stability, and the cokernel sheaf Q must have rank 1 and trivial determinant. We claim that Q = I Z for some scheme Z which necessarily has length n. This is clear if Q is torsion-free. However, if Q had torsion T , then there would be a torsion-free quotient Q tf of rank 1, , for a subscheme U.
Comparing determinants, we find D = det T effective so that D · H ≥ 0. This however contradicts stability of W → Q tf unless D · H = 0, in which case D = qE for some q ≥ 0. We argue that q = 0. Indeed, consider the kernel K of the surjection Using K ֒→ W and W is H-stable, it follows that K is H-stable as well. Write m = length (U ).
A direct calculation shows that χ(K, K) = χ(W − I U (−qE), W − I U (−qE)) = 2 + 4m + 4n(q − 1) + 3q 2 > 2, which contradicts stability. The only exception is The torsion part T must have trivial determinant hence it must be supported on points. Computing Euler characteristics This is only possible if T = 0 so that Q is torsion free, in fact Q = I U for some zero dimensional scheme U of length n. Thus we have identified the degeneracy locus D as It is straightforward to carry out the above argument in families. The crux of the matter is that (34) has constant rank, hence cohomology and base change commute over the degeneracy locus D. We set Since H 2 (V ⊗ W ⊗ E ∨ ) = 0 by Serre duality and stability, it follows from (33) that On D × S we have a universal exact sequence which restricts to (32) for each point of D. Here V, W, E are pulled back to the product from S, O(1) is the hyperplane bundle on D, and I Z is the restriction to D × S of the universal ideal sheaf on S [n] × S.
The Segre integral is calculated as an excess intersection on the degeneracy locus D as follows. Consider the Grassmannian G(2n, 4n − 1) of subspaces Write E for the tautological bundle, and let F denote the tautological quotient. The vector bundle has a natural section s obtained as the composition be the zero locus of this section. It consists of those pairs (Z, Λ) where In particular, h 0 (V ⊗ I Z ) ≥ dim Λ = 2n. Note now that while H 2 (V ⊗ I Z ) = 0 by Serre duality and stability. For (Z, Λ) in the degeneracy locus D 0 , we have then This shows that Z must be in the degeneracy locus D, and in this case h 1 (V ⊗ I Z ) = 1 by (34). We are thus forced to have Λ = H 0 (V ⊗ I Z ) so that Note that the projection onto S [n] induces an isomorphism D 0 ≃ D.
By [F], Section 14.4, we have where the excess virtual bundle E is with N the normal bundle of D 0 ⊂ S [n] × G(2n, 4n − 1). We have in K-theory, Putting all together, It remains to identify E explicitly in terms of the universal sequence (35). For simplicity we write identities in K-theory on fibers. To start, note that on D 0 we have We obtain Noting from (35) that it follows that . Here all tensor products are derived. We therefore finally calculate T S [n] = Ext 1 π (I Z , I Z ) (39) This last equality uses the fact that V, W are stable and rigid, and that V ⊗ W has no higher cohomology.
Lemma 5. Let n = 3ℓ. Assume where χ(V ) = 4n−1 and V 0 , W 0 are two rigid bundles satisfing conditions (21) and (31). We have Proof. The argument is an application of Reider's method. Assume that We construct a nontrivial extension The middle sheaf F is H-stable by an argument similar to that of Proposition 2. Indeed, if F is not H-stable, let G ֒→ F be the maximal semistable destabilizing subsheaf. Then rk G < rk F and µ H (G) ≥ µ H (F ) > µ H (V ∨ ⊗ E) .
We see that G cannot be a subsheaf of the kernel V ∨ ⊗ E since this would contradict the H-stability of the latter. Therefore, we have a nonzero morphism φ : G → W =⇒ µ H (G) < µ H (W ).

4.4.
Proof of Theorem 2. We now prove Theorem 2. The statement will follow combinatorially using the geometric input provided by Theorem 4.
Using (1) for the vector bundle V , we obtain that .
Combined with (40) we obtain the expression for A 4 claimed in Theorem 2.