Proper moduli spaces of isolated non-normal singularities

Abstract

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.

26 pages. Comments welcome