Ideals of the cohomology rings of Hilbert schemes and their applications

2002/08/31 by Wei-Ping Li, Zhenbo Qin, Weiqiang Wang
Mathematics · #math.AG #math.QA #msc:14F25

paper · pdf

published as Trans. AMS, Vol.356 No.1 (2004), p245-265 · Final version to appear on Trans. AMS

arxiv created 2003/10/23 · arxiv updated 2009/11/30

Abstract

We study the ideals of the rational cohomology ring of the Hilbert scheme X^[n] of n points on a smooth projective surface X. As an application, for a large class of smooth quasi-projective surfaces X, we show that every cup product structure constant of H^*(X^[n]) is independent of n; moreover, we obtain two sets of ring generators for the cohomology ring H^*(X^[n]). Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between H^*(X^[n], C) and H^*_orb(X^[n]/S_n, C) for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.

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