Lehn's conjecture for Segre classes on Hilbert schemes of points of surfaces and generalizations Lecture 2 : Hilbert schemes of log surfaces. Rahul Pandharipande (1) Let S be a nonsingular projective surface and let D in S be a nonsingular projective curve. Let S/D[n] be the Hilbert scheme of points of a surface S relative to D. (i) For the usual Hilbert scheme of point S[n], the supports move freely in S. For S/D[n] the space bubbles when the supports head to the relative divisor D. (ii) While S[n] is very familiar, the log geometry S/D[n] has been much less studied. (iii) For those familiar with relative DT, stable pairs, etc. S/D[n] is a special case of the construction. (iv) For computation of the Betti numbers of S/D[n], see the Princeton Ph.D. thesis of I. Setayesh. (2) The main questions and most of the discussion of Lecture 1 can be carried out for the relative geometry S/D. An important example is S = P1 x C with D = infty x C . Then S/D carries a T= C* x C* action extending the torus action on C^2 = S - D. Here, T acts with weight s,t at the origin (0,0) and with weights -s,t at (infty,0). Let T also act on the trivial line bundle L = O_{S} -> S equivariantly with weight w. We will now study the partition function Z_{S/D}(q,s,t,w) = \sum_{n=0}^{infty} q^n \int_{S/D[n]} Seg(L[n]). which nicely factors as Z_{S/D} = Z_vert * Z_rel where Z_vert is the localization vertex at (0,0) from Lecture 1 and Z_rel is the localization vertex associated to the rubber at (infty,0). Why is the relative geometry S/D here better than the the naked vertex Z_vert from Lecture 1? Answer: S/D is compact in the s direction related to the first C* factor of T => the q-coefficients of Z_{S/D} are polynomials in s, while the q-coefficients of Z_vert have denominators in s,t, To use the polynomality of Z_{S/D}, we must of study the series Z_rel via the rubber calculus. See [MNOP2] for a parallel study in the case of DT theory of 3-folds. (3) Rubber calculus The T fixed locus of S/D[n] with all points over infty in P1 is not isolated. Rather, the T-fixed locus is the rubber moduli space R[n] of dimension 2n-1. The rubber space R[n] is the Hilbert scheme of points of P1 x C relative to both 0 x C and infty x C (with bubbling when the the supports meet either) up to the C*-scale. It is the C*-scale equivalence which brings the dimension down to 2n-1. Localization gives the formula Z_rel = 1 + \sum_{n=1}^{infty} q^n \int_{R[n]} Seg(L[n]) / (-s-Psi) where L[n] is the tautological bundle on the rubber and -Psi is the tangent line associated to the relative divisor. We can expand Z_rel as a series in s: Z_rel = 1 + Z_1/(-s) + Z_2/(-s)^2 + Z_3/(-s)^3 + ... . The crucial equation of the rubber calculus is D Z_k = Z_{k-1} * D Z_1 for k>=2 . Some comments (i) The operator D is defined as D = q d/dq. (ii) The equation comes from producing a section of the cotangent line Psi. The solution to the rubber differential equation is Z_k = Z_1^k/k! so we have Z_rel = exp (- Z_1 / s ) . And what is Z_1 ? By definition, Z_1 is exactly the integral of Seg(L[n]) over R[n], Z_1 = \sum_{n=1}^{infty} q^n \int_{R[n]} Seg(L[n]) . Since the first C* factor of T does not act on R[n], Z_1 has no s-dependence (Z_1 depends only on t,w). Since w acts only on the line bundle L, Z_1 is polynomial in w. Conclusion: Log Z_rel = - Z_1/ s and has exactly simple poles in s and no poles in w. (4) Vertex analysis By geometry => the q-coefficients of Z_{S/D} are polynomial in s => the q-coefficients of Log Z_{S/D} are polynomial in s => since Log Z_{S/D} = Log Z_vert + Log Z_rel, the q-coefficients of Log Z_vert must have at most simple poles in s (and no poles in w). => since Log Z_vert is symmetric in s,t, the q-coefficients of Log Z_vert must have at most simple poles in t. => We can write F_0 F_1 F_2 Log(Z_vert)= ------ + ----- + ------ + ... . st st st The geometric proof of Theorem 2 is now complete. (5) Further comments. (a) We can calculate Z_1/s as exactly the polar part of Log(Z_vert). Since we can calculate Log(Z_vert) by localization, we have a method of computing \int_{R[n]} Seg(L[n]). (b) Since Log(Z_vert) has at most simple poles in t, \int_{R[n]} Seg(L[n]) must also have at most simple poles in t. We conclude (i) \int_{R[n]} Seg_{<2n-2}(L[n]) = 0, (ii) \sum_{n=1}^{infty} q^n \int_{R[n]} Seg_{2n-2}(L[n]) = F_0, where F_0 is determined in Theorem 4 of Lecture 1, An interesting question is to find a direct geometric approach to (i), (ii), and the computation of the higher Segre classes over rubber. (6) Compact surfaces Let S be a nonsingular projective toric surface with a T-action. Using localization, Theorem 2 immediately implies Theorem 3 of Lecture 1 for the T-equivariant theory of S. In fact, using further localization analysis, we can prove the corresponding statement for relative geometries in the toric case. Let S be a nonsingular projective toric surface, and let D be a nonsingular toric divisor. Then, Theorem 3 holds for the T-equivariant theory of S/D with the rules (s+t) => c1(S/D) st => c2(S/D) w => c1(L). Here c1(S/D) and c2(S/D) are the Chern classes of log tangent bundle of S/D --- dual to the bundle of holomorphic differentials with log poles along D. (7) Sketch of Proof of Theorem 3 in all cases. (a) We need Theorem 1 for families of log surfaces. The statement is unchanged (with the relative tangent bundle of the family replaced by the relative log tangent bundle of the log family). (b) Using (a), we can determine the universal polynomials of Theorem 1 using families of log surfaces. (c) While Theorem 3 for the T-equivariant theory of nonsingular projective toric surfaces was not enough to determine the universal polynomials, Theorem 3 for the T-equivariant theory of nonsingular projective log toric surfaces is enough. (8) A fixed surface S Let S be a fixed nonsingular projective surface with a nonsingular divisor D. Via Theorem 3, we have Z_S(q) = Exp( A1 * c1^2 + A2 * c2 + A3 * c1*L + A4 * L^2 ). So the Segre integral question amounts exactly to determining the 4 q-series A1, A2, A3, A4. Also Z_{S/D}(q) = Exp( A1 * c1(S/D)^2 + A2 * c2(S/D) + A3 * c1(S/D)*L + A4 * L^2 ) for the same functions. What are these functions? We will see in Lecture 3.