Moduli of $\mathbb{Q}$-Gorenstein pairs and applications

TL;DR

This paper develops a framework for constructing proper moduli spaces of $Q$-Gorenstein pairs, ensuring projectivity and avoiding certain pathologies, with applications to the moduli space of stable pairs.

Contribution

It introduces the notion of $Q$-stable pairs and demonstrates their role in creating proper, projective moduli spaces, improving understanding of moduli of pairs.

Findings

Proper moduli space with projective coarse moduli space constructed

$Q$-stable pairs prevent pathologies in moduli spaces

Alternative proof of projectivity for moduli space of stable pairs

Abstract

We develop a framework to construct moduli spaces of $\mathbb{Q}$-Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of $\mathbb{Q}$-stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than $\frac{1}{2}$. Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.