Resolution of Singularities of Arithmetical Threefolds I

Vincent Cossart (1) , Olivier Piltant (1)

1 LMV - Laboratoire de Mathématiques de Versailles

This article contains the first and main part of the proof of the Resolution of Singularities Conjecture for Arithmetical Threefolds (theorem 1.1). This result applies in particular to integer models of projective surfaces over number fields or over complete discrete valuation rings and produces an everywhere regular integer projective model (corollary 1.2). The text contains material which is valid in all dimensions and a reduction of theorem 1.1 to a projection theorem 4.4. The proof of theorem 4.4 is postponed to "Resolution of Singularities of Arithmetical Threefolds II" (to be posted on this website) and is adapted from the equicharacteristic techniques of [21] chapters 2,3 and 4.

Mots clés: resolution of singularities; Abhyankar; valuations; blowing up; integer models

Domaines: Géométrie algébrique [math.AG]

Vincent Cossart, Olivier Piltant. Resolution of Singularities of Arithmetical Threefolds I. 2013. ⟨hal-00873967⟩