ABHYANKAR 82nd BIRTHDAY CONFERENCE
SPEAKER: Vincent Cossart
TITLE: Desingularization in dimension~3, mixed characteristic
ABSTRACT: Joint work with Olivier Piltant. The main Theorem is the following.
MAIN THEOREM.
Let $C$ be an regular excellent curve with function field $F$.
Let $S/F$ be a reduced algebraic projective surface and $\X$ be a flat
projective $C$-scheme with generic fiber $\X_F=S$.
Then there exists a projective birational morphism $\pi : \X'\fleche \X$ with the following properties:
$\bullet$ $\X'$ is everywhere regular.
$\bullet$ $\pi$ induces an isomorphism $\pi^{-1}(\Reg(\X)) \iso \Reg(\X)$.
$\bullet$ $\pi^{-1}(\Sing(\X))$ is a normal crossings divisor on $\X'$.
In this conference we would like to focus on the uniformization problem.
We want to explain the main step in reducing the local uniformization
problem to the -already solved- uniformization problem in
characteristic $p>0$ for Artin-Schreier or purely inseparable extensions
of degree $p$ over a regular ring of dimension~3.