Stable varieties with a twist

TL;DR

This paper introduces a novel stack-based approach to defining the moduli of stable varieties, ensuring invariants remain constant in families and simplifying the construction of the moduli stack.

Contribution

It proposes adding natural stack structures to stable varieties, preserving key invariants and providing a clear construction of the moduli stack as a global quotient.

Findings

Stack structures preserve numerical invariants in deformations.

Moduli stack constructed as a global quotient of a Hilbert scheme.

Approach clarifies the family behavior of stable varieties.

Abstract

We describe a new approach to the definition of the moduli functor of stable varieties. While there is wide agreement as to what classes of varieties should appear, the notion of a family of stable surfaces is quite subtle, as key numerical invariants may fail to be constant in flat families. Our approach is to add natural stack structure to stable varieties. For example, given a canonical model we take the global-quotient stack structure arising from its realization as Proj of the canonical ring. Deformations of the stack structure preserve key numerical invariants of the stable variety, including the top self-intersection of the canonical divisor. This approach yields a transparent construction of the moduli stack of stable varieties as a global quotient of a suitable Hilbert scheme of weighted projective stacks.