Seminar on de Jong's alteration theorem
Friday 1012, M 311 and on Zoom.
Program
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, semistability 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 
DenisCharles Cisinski 
Introduction 
28.10.22 
Robin Carlier 
Preparation: semistable 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 threepoint lemma 
02.12.22 
Niklas Kipp 
Hilbert schemes 
09.12.22 
RitheeshKrishna 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 

Semistable curves over a DVR 
03.02.23 

Resolving semistable curves over a regular base 
10.02.23 
Niklas Kipp 
Controling the étale locus of the alteration 