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September 2011 Elimination algebras and inductive arguments in resolution of singularities
Ana Bravo, Orlando E. Villamayor
Asian J. Math. 15(3): 321-356 (September 2011).

Abstract

Over fields of characteristic zero, resolution of singularities is achieved by means of an inductive argument, which is sustained on the existence of the so called hypersurfaces of maximal contact. We report here on an alternative approach which replaces hypersurfaces of maximal contact by generic projections. Projections can be defined in arbitrary characteristic, and this approach has led to new invariants when applied to the open problem of resolution of singularities over arbitrary fields. We show here how projections lead to a form of elimination of one variable using invariants that, to some extent, generalize the notion of discriminant.

This exposition draws special attention on this form of elimination, on its motivation, and its use as an alternative approach to inductive arguments in resolution of singularities. Using techniques of projections and elimination one can also recover some well known results. We illustrate this by showing that the Hilbert-Samuel stratum of a d-dimensional non-smooth variety can be described with equations involving at most d variables.

In addition this alternative approach, when applied over fields of characteristic zero, provides a conceptual simplification of the theorem of resolution of singularities as it trivializes the globalization of local invariants.

Citation

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Ana Bravo. Orlando E. Villamayor. "Elimination algebras and inductive arguments in resolution of singularities." Asian J. Math. 15 (3) 321 - 356, September 2011.

Information

Published: September 2011
First available in Project Euclid: 28 February 2012

zbMATH: 1360.14046
MathSciNet: MR2838211

Subjects:
Primary: 14E15

Keywords: Rees algebras , resolution of singularities

Rights: Copyright © 2011 International Press of Boston

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Vol.15 • No. 3 • September 2011
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