发表机构
University of Washington（华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解答了Dan Abramovich关于特定态射是否为加权爆破的问题，证实了两类情形下的结论，并将其应用于Hassett带权稳定曲线模叠的约化态射判定。
AI 中文摘要
本文对Dan Abramovich提出的问题给出了部分解答：考虑光滑分离Deligne–Mumford stacks间具有连通纤维的真态射 $f: \boldsymbol{\textit{X}} \to \boldsymbol{\textit{Z}}$，该态射在 $\boldsymbol{\textit{X}}$ 的光滑有效Cartier除子 $\boldsymbol{\textit{E}} \boldsymbol{\textit{E}}$ 之外定义同构。那么，$f$ 是否为加权爆破？我们证实，当 $\boldsymbol{\textit{X}}$ 和 $\boldsymbol{\textit{Z}}$ 是 $\boldsymbol{\textit{C}}$ 上有限型的光滑分离概型，且 $f$ 是光滑分离Deligne–Mumford stacks的可表态射时，$f$ 是普通光滑爆破。进一步证明，当 $\boldsymbol{\textit{X}}$ 和 $\boldsymbol{\textit{Z}}$ 是光滑分离Deligne–Mumford曲面（即 $\boldsymbol{\textit{dim}} \boldsymbol{\textit{X}} = \boldsymbol{\textit{dim}} \boldsymbol{\textit{Z}} = 2$）时，$f$ 是加权爆破。作为应用，我们确定了Hassett带权稳定曲线模叠间的约化态射何时由沿光滑中心的爆破给出。
英文摘要
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.
Comments32 pages. Comments welcome!
