Symmetric products of a semistable degeneration of surfaces

TL;DR

This paper constructs explicit models of symmetric products of surface degenerations using toric geometry, computes their stringy E-polynomials, and develops minimal models of Hilbert scheme degenerations.

Contribution

It introduces explicit toric models for symmetric products and Hilbert schemes of surface degenerations, advancing understanding of their geometric and stringy invariants.

Findings

Constructed a $V$-normal crossing Gorenstein canonical model.

Calculated the stringy $E$-polynomial of the relative symmetric product.

Explicitly constructed a minimal model of degeneration of Hilbert schemes.

Abstract

We explicitly construct a $V$-normal crossing Gorenstein canonical model of the relative symmetric products of a local semistable degeneration of surfaces without a triple point by means of toric geometry. Using this model, we calculate the stringy $E$-polynomial of the relative symmetric product. We also construct a minimal model of degeneration of Hilbert schemes explicitly.