Seminar on de Jong's alteration theorem

Friday 10-12, M 311 and on Zoom.

The goal of the seminar is to review de Jong's alteration theorem: the approximation of any integral, separated scheme of finite type over a field by a regular one, turning any prescribed proper closed subscheme into the support of a normal crossings divisor. This is a fundamental theorem which is almost as good as Hironaka's resolution of singularities which is very useful to prove finiteness theorems in algebraic geometry, for instance (e.g. finiteness of ranks of étale cohomology groups, Künneth formulas without properness assumptions, Hodge theory...). The original source is de Jong's paper Smoothness, semi-stability and alterations which relies on the work of Delligne and Mumford on the moduli stack of stable curves with marked points in their paper The irreducibility of the space of curves of given genus. A more general theorem is proved in another paper of de Jong, Families of curves and alterations. We will essentially follow Brian Conrad's lecture notes, and possibly spend some time on a refined version by Bhatt and Snowden. The first talk (Introduction) will speak of resolution of singularities and give example of statements which use (variants of) resolution of singularities in their proof. The next 5 talks are about introducing the main geometric concepts needed to state and to begin to implement the proof of de Jong's theorem; the level of difficulty improves with time, but they all only deal with rather classical algebraic geometry. There are 4 talks on the construction of the moduli stack of stable curves, following Deligne and Mumford, and then 2 talks to finish the proof of de Jong's theorem. The remaining talks are devoted to variations of the theorem (resolving stable curves over a regular base) and the refinements of Bhatt and Snowden, proving for instance that every smooth variety over a perfect field is étale locally the complement of a normal crossings divisor in a smooth and projective one.
Date Speaker Topic
21.10.22 Denis-Charles Cisinski Introduction
28.10.22 Robin Carlier Preparation: semi-stable curves, (quasi)-excellent schemes
04.11.22 Sebastian Wolf First reductions
11.11.22 Johannes Gloßner Construction of good relative curves 1
18.11.22 Andrea Panontin Construction of good relative curves 2
25.11.22 Massimo Pippi The three-point lemma
02.12.22 Niklas Kipp Hilbert schemes
09.12.22 Ritheesh-Krishna Thiruppathi Stable curves
16.12.22 Pavel Sechin Deformations of stable curves
23.12.22 Suraj Yadav The stack of stable curves
13.01.23 Benedikt Preis Application of moduli stacks 1
20.01.23 Suraj Yadav Application of moduli stacks 2
27.01.23 Semi-stable curves over a DVR
03.02.23 Resolving semi-stable curves over a regular base
10.02.23 Niklas Kipp Controling the étale locus of the alteration