Coefficient and elimination algebras in resolution of singularities (original) (raw)
2011, Asian Journal of Mathematics
Given a variety X over a field k one wants to find a desingularization, which is a proper and birational morphism X ′ → X, where X ′ is a regular variety and the morphism is an isomorphism over the regular points of X. If X is embedded in a regular variety W , there is a notion of embedded desingularization and related to this is the notion of log-resolution of ideals in O W . When the field k has characteristic zero it is well known that the problem of resolution is solved. The first proof of the existence of resolution of singularities is due to H. Hironaka in his monumental work [Hir64] (see also ). If characteristic of k is positive the problem of resolution in arbitrary dimension is still open. See [Hau10] for recent advances and obstructions (see also [Hau03]). The proof by Hironaka is existential. There are constructive proofs, always in characteristic zero case, see for instance [VU89], [VU92], [BM97], we refer to [Hau03] for a complete list of references. Those constructive proofs give rise to algorithmic resolution of singularities, that allows to perform implementation at the computer [BS00], [FKP04].
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