Stable maps to quotient stacks with a properly stable point
Andrea Di Lorenzo, Giovanni Inchiostro
TL;DR
The paper develops a broad, stack-theoretic approach to compactifying the moduli of maps from curves to quotient stacks with projective good moduli spaces, by enlarging the target using extended weighted blow-ups to ensure an open Deligne–Mumford locus that is proper. Central to the method is a three-pronged strategy: (i) extend the target to a DM-open substack via Kirwan desingularization, (ii) leverage relative GIT and Luna slice techniques to control semistability and local structure, and (iii) implement extended weighted blow-ups to realize modular enlargements with preserved good moduli spaces. The authors prove a general existence theorem for proper DM quasimaps to the enlarged targets and establish boundedness and obstruction theories, enabling modular compactifications with boundary described by stable quasimaps to the enlargement. They apply the framework to moduli of fibered Calabi–Yau pairs, toric quotients, and GIT moduli spaces of plane cubics, and also prove Hassett-type results via a modular extended blow-up perspective. Overall, the work provides a versatile toolkit linking birational geometry, GIT, and moduli theory to produce robust, modular compactifications for maps to complex stacks and their quotients.
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.