Resolution of singularities of threefolds in positive characteristic I

Vincent Cossart (1) , Olivier Piltant (1)

1 LMV - Laboratoire de Mathématiques de Versailles

The purpose of this article and of "Resolution of singularities of threefolds in positive characteristic II" is to prove the theorem of resolution: resolution of singularities holds for algebraic varieties of dimension three over a field $k$ of characteristic $p>0$ whenever $k$ is differentially finite over some perfect subfield $k_0$. This condition is satisfied in particular when $k$ is a function field over $k_0$. The resolution of singularities $\pi: \tilde{Z}\rightarrow Z$ which we obtain is projective, birational and an isomorphism away from the singular locus of any given variety $Z$. It should be emphasized however that our construction of $\pi$ is purely existential: it neither respects embeddings of $Z$ in a regular space, nor is given by any resolution algorithm.

Mots clés: desingularisation; caracteristique positive

Domaines: Géométrie algébrique [math.AG]

Vincent Cossart, Olivier Piltant. Resolution of singularities of threefolds in positive characteristic I. Journal of Algebra, 2008, 320 (7), pp.1051-1082. ⟨hal-00139124⟩