Resolution of Singularities - Resolution For Schemes and Status of The Problem

Resolution For Schemes and Status of The Problem

It is easy to extend the definition of resolution to all schemes. Not all schemes have resolutions of their singularities: Grothendieck (1965, section 7.9) showed that if a locally Noetherian scheme X has the property that one can resolve the singularities of any finite integral scheme over X, then X must be quasi-excellent. Grothendieck also suggested that the converse might hold: in other words, if a locally Noetherian scheme X is reduced and quasi excellent, then it is possible to resolve its singularities. When X is defined over a field of characteristic 0, this follows from Hironaka's theorem, and when X has dimension at most 2 it was prove by Lipman. In general it would follow if it is possible to resolve the singularities of all integral complete local rings.

Hauser (2010) gave a survey of work on the unsolved characteristic p resolution problem.

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