Characterizing Contractions and Weighted Blowdowns

Abstract: 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.