Stable maps to quotient stacks with a properly stable point

Abstract

We compactify moduli stacks of maps from curves to a large class of quotient stacks $[W/G]$ admitting projective good moduli spaces. Our construction extends quasimap-theoretic compactifications and relies on a new birational operation for algebraic stacks, which we call an extended weighted blow-up. As applications, we construct compactifications of certain moduli spaces of fibrations in log Calabi-Yau pairs, of maps to stacks of the form $[W/{\mathbb{G}}_m^n]$, of maps to the GIT moduli stack of $2n$ points on ${\mathbb{P}}^1$, and of maps to the GIT moduli stack of plane cubics. In the appendix, we use extended weighted blow-ups to give a modular proof of a conjecture of Hassett.

arXiv commentv3: discussion of $\epsilon$-stability added, together with references for further applications of our main result. Proofs shortened, exposition improved