L'uniformisation locale des surfaces d'Artin-Schreier en caracteristique positive

Abstract : This thesis deals with uniformization, in characteristic p>0, of a rational valuation, in special cases where this valuation is centered on a singularity locally defined by the following equations :

- either z^p+f(x,y)=0, with f not a p-th power, and ordf >p,

- or z^p+e(x,y)z+f(x,y)=0, with ord (ez+f)>p (Artin-Schreier's case).

Historically, it was in such cases that all difficulty of resolving surfaces in positive characteristic was concentrated.


The novelty bringed in this work consists first in giving a bound to the
minimum number of closed point's blowing-ups needed to uniformize, and second
in anticipating (from the first ring) the Newton polygon's evolution and the
parameter's choice for the successive blowing-ups along the valuation.

In a first part, we come back on the Giraud's normal form of f in O_X(X)$
where X is a two dimensional regular scheme of characteristic p. The starting
point is an polynomial expansion of f with a generating sequence for the
valuation. We can then study and anticipate the behavior of this expansion and the associated Newton polygon modulo a p-th power. We then give a bound on the maximum number of blowing-ups needed for this polygon to become minimal, with only one vertex, and of maximal height one. This case correspond to the normal form of f.

In a second part, using this results for the two above-mentionned cases, we
give an algorithm witch anticipate, in the first ring, the translations on z
needed to keep a minimal Newton polygon during the blowing-ups seque ce (along
the valuation), and we quantify the maximal size of such a sequence with last ring corresponding to a quasi-ordinary singularity.
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Submitted on : Friday, December 6, 2002 - 6:26:08 PM
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Raphael Astier. L'uniformisation locale des surfaces d'Artin-Schreier en caracteristique positive. Mathématiques [math]. Université de Versailles-Saint Quentin en Yvelines, 2002. Français. ⟨tel-00002087⟩

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