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arXiv · 1808.04431

Exact Formulas for Invariants of Hilbert Schemes

Nate Gillman·Xavier Gonzalez·Matthew Schoenbauer
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Abstract

A theorem of Göttsche establishes a connection between cohomological invariants of a complex projective surface $S$ and corresponding invariants of the Hilbert scheme of $n$ points on $S.$ This relationship is encoded in certain infinite product $q$-series which are essentially modular forms. Here we make use of the circle method to arrive at exact formulas for certain specializations of these $q$-series, yielding convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the $q$-series' coefficients.

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Nate Gillman, Xavier Gonzalez, Matthew Schoenbauer. 2018-09-04. Exact Formulas for Invariants of Hilbert Schemes. https://doi.org/10.1007/s40993-018-0132-z

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