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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.

In dimension 3, the desingularization problem is reduced to local uniformization on Artin-Schreier and purely inseparable coverings of regular space.