Title: Intersection numbers on relative Hilbert scheme of points on surfaces
Speaker: Artan Sheshmani (OSU)
Seminar URL: https://research.math.osu.edu/reps/
Abstract: In the remarkable work of Nakajima, it was shown that the study of infinite dimensional Heisenberg algebra over the homology group of moduli space of torsion free rank 1 sheaves on a surface S and certain operators, known as Nakajima operators, given by the generators of this Heisenberg algebra, provides a tool to study the cohomology of Hilb^n(S) (the absolute Hilbert scheme of n points on S). Later Okounkov and Carlsson generalized the work of Nakajima and constructed a rather different set of operators acting on homology groups of Hilb^n(S). These operators, known as "vertex operators", depend on choice of a fixed line bundle, M, over S, and could be explicitly written with respect to the Nakajima operators. In this talk, I will talk about joint work with Amin Gholampour on obtaining the "relative version" of Okounkov-Carlsson generating series for certain top intersection numbers of "relative" Hilbert schemes of points on a surface relative to an effective smooth divisor. We find a formula relating these numbers to the corresponding intersection numbers on the non-relative Hilbert schemes. In particular, we obtain a relative version of the explicit formula found by Carlsson-Okounkov for the Euler class of the twisted tangent bundle of Hilbert schemes. We also prove a relative version of (conjectural) generating series of the top Segre class of tautological bundles associated to a line bundle on S. These integrals were studied by Lehn, as well as Donaldson, in connection with the computation of instanton invariants.
