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

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{\mathscr{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.