Désingularisation en dimension 3, caractéristique mixte

Vincent Cossart

13 septembre 2013

Conférence dédiée à Shreeram Shankar Abhyankar, 1930-2012.

Travail en commun avec Olivier Piltant.

Theorem 1. (Cossart-Piltant) Let $C$ be an integral Noetherian curve which is excellent and ${\cal X}/C$ be a reduced and separated scheme of finite type and dimension at most three. There exists a proper birational morphism $\pi : \ {\cal X}' \rightarrow {\cal X}$ with the following properties:

(i)
${\cal X}'$ is everywhere regular;
(ii)
$\pi$ induces an isomorphism $\pi^{-1}(\mathrm{Reg}({\cal X})) \simeq \mathrm{Reg}({\cal X})$;
(iii)
$\pi^{-1}(\mathrm{Sing}({\cal X}))$ is a normal crossings divisor on ${\cal X}'$.
If furthermore ${\cal X}\backslash \mathrm{Sing}{\cal X}$ is quasi-projective, one may furthermore take ${\cal X}'$ projective.

Par une réduction "à la Abhyankar" [1], le théorème ci-dessus est une conséquence du théorème suivant :

Theorem 2. (Cossart-Piltant) Let $(S,m_S,k)$ be an excellent regular local ring of dimension three, quotient field $K:=QF(S)$ and residue characteristic $\mathrm{char}k=p>0$. Let

\begin{displaymath}
h:=X^p+f_1X^{p-1}+ \cdots +f_p \in S[X], \ f_1, \ldots , f_p \in S
\end{displaymath} (1)

be a reduced polynomial, ${\cal X} :=\mathrm{Spec}(S[X]/(h))$ and $L:=\mathrm{Tot}(S[X]/(h))$ be its total quotient ring. Assume that $h$ satisfies one of the following assumptions:
(i)
${\cal X}$ is $G$-invariant, where $\mathrm{Aut}_K(L)=\mathbf{Z}/p =:G$, or
(ii)
$\mathrm{char}K=p$ and $f_1= \cdots =f_{p-1}=0$.
Let $\mu$ be a valuation of $L$ which is centered in $m_S$. There exists a composition of local Hironaka-permissible blowing ups:
\begin{displaymath}
({\cal X}=:{\cal X}_0,x_0) \leftarrow ({\cal X}_1,x_1) \leftarrow \cdots \leftarrow ({\cal X}_r,x_r),
\end{displaymath} (2)

where $x_i \in {\cal X}_i$ is the center of $\mu$, such that $({\cal X}_r,x_r)$ is regular.

Le cas (ii) est déjà résolu [2]. Dans cet exposé, nous allons expliciter le cas (i) à l'aide de la théorie des polyèdres d'Hironaka et des gradués associés : si le discriminant de $h$ est monomial, les formes initiales de $h$ pour les valuations correspondant aux faces du polyèdre sont alors d'Artin-Schreier ou purement inséparables. C'est le point clef de notre preuve.

La preuve complète du théorème de désingularisation sera exposée du 1 au 11 octobre à Ratisbonne.

http://tinyurl.com/CPschool13


Bibliography

1
COSSART V., PILTANT O., Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings, J. Algebra 320 (2008), no. 3, 1051-1082.

2
COSSART V., PILTANT O., Resolution of singularities of threefolds in positive characteristic II, J. Algebra 321 (2009), no. 7, 1836-1976.

3
COSSART V., PILTANT O., Characteristic polyhedra of singularities without completion, preprint arXiv:1203.2484 (2012), 1-6.

4
COSSART V., PILTANT O., Resolution of Singularities of Threefolds in Mixed Characteristics. Case of small multiplicity, to appear in RACSAM (2013), 1-39.