Exact Formulas for Invariants of Hilbert Schemes
arXiv:1808.04431 · doi:10.1007/s40993-018-0132-z
Abstract
A theorem of Göttsche establishes a connection between cohomological invariants of a complex projective surface and corresponding invariants of the Hilbert scheme of points on This relationship is encoded in certain infinite product -series which are essentially modular forms. Here we make use of the circle method to arrive at exact formulas for certain specializations of these -series, yielding convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the -series' coefficients.
25 pages, 1 figure, Accepted for publication in Research in Number Theory