Desingularization of quasi-excellent schemes in characteristic zero

By M. Temkin. 2007

153 citations recorded by the source corpus.

Grothendieck proved in EGA IV that if any integral scheme of finite type over a locally noetherian scheme X admits a desingularization, then X is quasi-excellent, and conjectured that the converse is probably true. We prove this conjecture for noetherian schemes of characteristic zero. Namely, starting with the resolution of singularities for algebraic varieties of characteristic zero, we prove the resolution of singularities for noetherian quasi-excellent Q-schemes.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8b213dca9dc67e9c2556d0ed45bed4cdcd2a4fac

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In Computational Theory and Mathematics

1. M. Mustaţă
2. Mattias Jonsson
3. S. Boucksom
4. C. Favre
5. Tommaso de Fernex
6. J. Kollár
7. Chi Li
8. L. Ein

In Geometry and Topology

1. Mattias Jonsson
2. M. Mustaţă
3. S. Boucksom
4. Chi Li
5. C. Favre
6. Tommaso de Fernex
7. H. Blum
8. Robert J. Berman

Citations in the corpus

• Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case (2009, methodology): “cable case of desingularization of general qe schemes over Q is when the schemes are of finite type over a ring k[[t1,...,tn]] with k a field. For example, this case (in its non-functorial version from [Tem1]) was used for studying log-canonical thresholds in [FEM], motivic integration in [Nic], and d”
• Shokurov's ACC conjecture for log canonical thresholds on smooth varieties (2009, background): “g canonical pairs in the setting of algebraic varieties, see [Kol1] or Chapter 9 in [Laz], while for the case of the spectrum of a formal power series ring we refer to [dFM]. The key point is that by [Tem], log resolutions exist also in the latter case, and therefore the usual theory of log canonica”
• Singular semipositive metrics in non-Archimedean geometry (2011, methodology): “g components of the possibly non-connected E J’s. Since khas characteristic zero, each S-variety is a Q-scheme, which is furthermore excellent since it has nite type over S. It therefore follows from [Tem06] that for any S-variety Xwith smooth generic ber there exists a vertical blow-up X0!Xsuch tha”
• Which powers of holomorphic functions are integrable (2008, background): “Indeed, resolution of singularities is known for complete local rings [Tem07] and then (11.”
• Functorial desingularization over Q: boundaries and the embedded case (2009, methodology): “rreducible components in the formulation and proof of [Tem08, Theorem 2.2.11]. Actually this was in the original argument I heard from Bierstone-Milman! The correction does not affect anything else in [Tem08]. (ii) Functorial semi-strict desingularization does not exist. Indeed, such desingularizatio”