Characterizing Contractions and Weighted Blowdowns

This paper gives a partial answer to a question of Dan Abramovich: consider a proper morphism $f:\mathcal{X}\to \mathcal{Z}$ with connected fibers, between smooth separated Deligne–Mumford stacks, which defines an isomorphism away from a smooth effective Cartier divisor $\mathcal{E}\subseteq \mathcal{X}$. Then, is $f$ a weighted blowup? We confirm that $f$ is an ordinary smooth blowup when $\mathcal{X}$ and $\mathcal{Z}$ are smooth separated schemes of finite type over $\mathbb{C}$, and when $f:\mathcal{X}\to \mathcal{Z}$ is a representable morphism of smooth separated Deligne–Mumford stacks. Further, we show that $f$ is a weighted blowup when $\mathcal{X}$ and $\mathcal{Z}$ are smooth separated Deligne–Mumford surfaces, i.e., $\dim \mathcal{X}=\dim \mathcal{Z}=2$. As an application we determine when a reduction morphism between Hassett moduli stacks of weighted stable curves is given by a blowup along a smooth center.