Simultaneous semi-stable reduction for curves with ADE singularities

TL;DR

This paper extends the concept of simultaneous resolution from algebraic surfaces to curves with ADE singularities, providing explicit resolutions and discussing their implications for moduli spaces.

Contribution

It introduces a natural resolution for families of ADE curves and explores the lifting of moduli maps to the stack, highlighting differences from surface cases.

Findings

Explicit resolution of the rational map to ar M_g

Discussion on lifting to the moduli stack ar al M_g

Weyl cover not sufficient for lifting in ADE case

Abstract

A key tool in the study of algebraic surfaces and their moduli is Brieskorn's simultaneous resolution for families of algebraic surfaces with simple (du Val or ADE) singularities. In this paper we show that a similar statement holds for families of curves with at worst simple (ADE) singularities. For a family $\mathscr X\to B$ of ADE curves, we give an explicit and natural resolution of the rational map $B\to \bar M_g$. Moreover, we discuss a lifting of this map to the moduli stack $ \bar {\mathcal M}_g$, i.e. a simultaneous semi-stable reduction for the family $\mathscr X/B$. In particular, we note that in contrast to what might be expected from the case of surfaces, the natural Weyl cover of $B$ is not a sufficient base change for a lifting of the map $B\to \bar M_g$ to $\bar {\mathcal M}_g$.