Resolution of singularities of threefolds in positive characteristic II

Abstract

Together with [Vincent Cossart, Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings, J. Algebra 320 (3) (2008) 1051-1082], this article gives a complete proof of desingularization of quasiprojective varieties of dimensional 3 on fields which are differentially finite over perfect fields.

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“…in positive characteristic [Cu,CP1,CP2]. Cutkosky reduces Abhyankar's proof (of over 500 pages) of resolution in characteristic greater than 5 to some forty pages, Cossart and Piltant establish the result with considerably more effort for arbitary fields.…”

“…Cossart and Piltant succeeded in removing the restriction on the characteristic and the algebraic closedness of the ground field. The resulting proof is rather long and challenging [CP1,CP2], based on ideas of [Co].…”

Formal prime ideals of infinite value and their algebraic resolution

“…Recently, there has been progress on local uniformization in positive characteristic, including the work of Cossart and Piltant [5], [6], Kuhlmann [14], Knaf and Kuhlmann [13], Spivakovsky [18], [17], Temkin [20] and Teissier [19]. Some recent progress on understanding valuations in the context of algebraic geometry has been made by Favre and Jonnson [10], Ghezzi, Hà and Kascheyeva [15], Vaquié [21] and others.…”

Elimination with Applications to Singularities in Positive Characteristic

“…We also refer to [8] and [7] for new proofs on non-embedded resolution of singularities of schemes of dimension 3 and positive characteristic.…”