Moduli spaces of hyperelliptic curves with A and D singularities

TL;DR

This paper constructs and studies moduli spaces of hyperelliptic curves with specific A and D singularities, exploring how stability conditions vary with parameters and connect different geometric structures.

Contribution

It introduces a new framework for moduli spaces of hyperelliptic covers with controlled singularities, interpolating between known stability conditions and deformation quotients.

Findings

Defined moduli spaces with A and D singularities

Established stability conditions depending on parameters

Connected different geometric moduli via parameter variation

Abstract

We introduce moduli spaces of quasi-admissible hyperelliptic covers with at worst A and D singularities. The stability conditions for these moduli spaces depend on two parameters describing allowable singularities. By varying these parameters, we interpolate between the stacks of stable limits of A and D singularities and the quotients of the versal deformation spaces of A and D singularities by a natural $\mathbb{G}_m$ action.