Stable maps to quotient stacks with a properly stable point

TL;DR

This paper develops a new birational transformation called extended weighted blow-up to compactify moduli stacks of maps to quotient stacks, enabling new applications in algebraic geometry including compactifications of various moduli spaces.

Contribution

It introduces the extended weighted blow-up, a novel birational transformation, and applies it to compactify moduli stacks of maps to quotient stacks with stable points.

Findings

Constructed a compact moduli stack for fibered log-Calabi-Yau pairs.

Provided a criterion for extended weighted blow-up of algebraic stacks.

Proved a modular version of Hassett's conjecture on weighted pointed rational curves.

Abstract

We compactify the moduli stack of maps from curves to certain quotient stacks $\mathcal{X}=[W/G]$ with a projective good moduli space, extending previous results from quasimap theory. For doing so, we introduce a new birational transformation for algebraic stacks, the extended weighted blow-up, to prove that any algebraic stack with a properly stable point can be enlarged so that it contains an open substack which is proper and Deligne-Mumford. As a first application, we use our main theorem to construct a compact moduli stack for certain fibered log-Calabi-Yau pairs. We further apply our result to construct a compactification of the space of maps to $\mathcal{X}$ when $\mathcal{X}$ is respectively: a quotient by a torus of a proper Deligne-Mumford stack; a GIT compactification of the stack of binary forms of degree $2n$; a GIT compactification of the stack of $2n$-marked smooth rational curves, and a GIT compactification of the stack of smooth plane cubics. In the appendix, we give a criterion for when a morphism of algebraic stacks is an extended weighted blow-up, and we use it in order to give a modular proof of a conjecture of Hassett on weighted pointed rational curves.