The Stacks project

55.1 Introduction

In this chapter we prove the semistable reduction theorem for curves. We will use the method of Artin and Winters from their paper [Artin-Winters].

It turns out that one can prove the semistable reduction theorem for curves without any results on desingularization. Namely, there is a way to establish the existence and projectivity of moduli of semistable curves using Geometric Invariant Theory (GIT) as developed by Mumford, see [GIT]. This method was championed by Gieseker who proved the full result in his lecture notes [Gieseker]1. This is quite an amazing feat: it seems somewhat counter intuitive that one can prove such a result without ever truly studying families of curves over a positive dimensional base.

Historically the first proof of the semistable reduction theorem for curves can be found in the paper [DM] by Deligne and Mumford. It proves the theorem by reducing the problem to the case of Abelian varieties which was already known at the time thanks to Grothendieck and others, see [SGA7-I] and [SGA7-II]). The semistable reduction theorem for abelian varieties uses the theory of Néron models which in turn rests on a treatment of birational group laws over a base.

The method in the paper by Artin and Winters relies on desingularization of singularities of surfaces to obtain regular models. Given the existence of regular models, the proof consists in analyzing the possibilities for the special fibre and concluding using an inequality for torsion in the Picard group of a $1$-dimensional scheme over a field. A similar argument can be found in a paper [Saito] of Saito who uses étale cohomology directly and who obtains a stronger result in that he can characterize semistable reduction in terms of the action of the inertia on $\ell $-adic étale cohomology.

A different approach one can use to prove the theorem is to use rigid analytic geometry techniques. Here we refer the reader to [vanderPut] and [Arzdorf-Wewers].

The paper [Temkin] by Temkin uses valuation theoretic techniques (and proves a lot more besides); also Appendix A of this paper gives a nice overview of the different proofs and the relationship with desingularizations of $2$ dimensional schemes.

Another overview paper that the reader may wish to consult is [Abbes-ssr] written by Ahmed Abbes.

[1] Gieseker's lecture notes are written over an algebraically closed field, but the same method works over $\mathbf{Z}$.

Comments (2)

Comment #3988 by Matthieu Romagny on

missing word in last sentence of 2nd paragraph : without ever truly studying families of curves over a positive dimensional base

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