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.
||Preparation: semi-stable curves, (quasi)-excellent schemes
||Construction of good relative curves 1
||Construction of good relative curves 2
||The three-point lemma
||Deformations of stable curves
||The stack of stable curves
||Application of moduli stacks 1
||Application of moduli stacks 2
||Semi-stable curves over a DVR
||Resolving semi-stable curves over a regular base
||Controling the étale locus of the alteration