A criterion for smooth weighted blow-downs (with an appendix by Stephen Obinna)

Abstract

We establish a criterion for determining when a smooth Deligne–Mumford stack is a weighted blow-up. More precisely, given a smooth Deligne–Mumford stack $\mathcal{X}$ and a Cartier divisor $\mathcal{E}\subset \mathcal{X}$ such that (1) $\mathcal{E}$ is a weighted projective bundle over a smooth Deligne–Mumford stack $\mathcal{Y}$ and (2) for every $y\in \mathcal{Y}$ we have ${\mathcal{O}}_{\mathcal{X}}(\mathcal{E}){|}_{{\mathcal{E}}_y}\simeq {\mathcal{O}}_{{\mathcal{E}}_y}(-1)$, there exists a contraction $\mathcal{X}\to \mathcal{Z}$ to a smooth Deligne–Mumford stack $\mathcal{Z}$. Moreover, the stack $\mathcal{X}$ can be recovered as a weighted blow-up along $\mathcal{Y}\subset \mathcal{Z}$ with exceptional divisor $\mathcal{E}$, and $\mathcal{Z}$ is a push-out in the category of algebraic stacks. As an application, we show that the moduli stack ${\overline{\mathcal{M}}}_{1,n}$ of stable $n$-pointed genus 1 curves is a weighted blow-up of the stack of pseudo-stable curves. A key step is a reconstruction result for smooth Deligne–Mumford stacks that may be of independent interest.