Inducing coverings on Hilbert schemes

TL;DR

This paper provides a geometric description of coverings of the Hilbert square of a surface and demonstrates that the Hilbert square of an irreducible symplectic surface is itself an irreducible symplectic variety.

Contribution

It introduces an explicit geometric framework for all coverings of Hilbert squares on certain surfaces and applies this to symplectic surfaces to establish their Hilbert squares as irreducible symplectic varieties.

Findings

Explicit geometric description of coverings of Hilbert squares.

Hilbert square of an irreducible symplectic surface is an irreducible symplectic variety.

Application of the construction to symplectic surfaces.

Abstract

We find an explicit geometric description of all coverings of the Hilbert square on a normal, complex, quasi-projective surface with finite fundamental group. We then apply this construction to show that if $\Sigma$ is an irreducible symplectic surface then its Hilbert square is an irreducible symplectic variety.