AI 中文总结
本文引入褶皱函子$\rm{Crimp}_S^d(X)$，验证Artin准则证明其对应代数空间性质，进而构造零亏格几何单枝曲线的半稳定条件与模空间，研究孤立非正规奇点的模空间。
AI 中文摘要
我们通过引入$(X \to S) $余秩为$d$的褶皱函子$\rm{Crimp}_S^d(X)$来研究孤立非正规奇点的模空间，该函子将Ishii的有限余长子环的模函子全局化。通过验证Artin准则，我们证明若$X \to S$是有限展示的分离态射，则$\rm{Crimp}_S^d(X)$由$S$上有限展示的分离代数空间表示，且当$X \to S$是真态射时该空间是真的。我们利用此构造零亏格几何单枝曲线的半稳定条件和模空间。
英文摘要
We study the moduli of isolated non-normal singularities by introducing the functor $\mathrm{Crimp}_S^d(X)$ of crimpings of $(X \to S)$ corank $d$. This globalizes Ishii's moduli functor of subrings of finite colength. By verifying Artin's criteria, we prove that if $X \to S$ is a separated morphism of finite presentation, then $\mathrm{Crimp}_S^d(X)$ is represented by a separated algebraic space of finite presentation over $S$, which is proper when $X \to S$ is proper. We use this to construct a semistability condition and moduli space for geometrically unibranch curves of genus zero.
Comments26 pages. Comments welcome
