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Monoidal transforms and invariants of singularities in positive characteristic

Part of: Birational geometry

Published online by Cambridge University Press:  19 June 2013

Angélica Benito  and
Orlando E. Villamayor U.
Angélica Benito
Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, USA email abenitos@umich.edu
Orlando E. Villamayor U.
Affiliation:
Dpto. Matemáticas, Universidad Autónoma de Madrid and ICMAT-UAM, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain email villamayor@uam.es
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Abstract

The problem of resolution of singularities in positive characteristic can be reformulated as follows: fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of $X$ be a finite composition of monoidal transformations with centers included in the $n$-fold points of $X$, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an $n$-sequence such that the final strict transform of $X$ has no points of multiplicity $n$ (no $n$-fold points). In characteristic zero, such an $n$-sequence is defined in two steps. The first consists of the transformation of $X$ to a hypersurface with $n$-fold points in the so-called monomial case. The second step consists of the elimination of these $n$-fold points (in the monomial case), which is achieved by a simple combinatorial procedure for choices of centers. The invariants treated in this work allow us to present a notion of strong monomial case which parallels that of monomial case in characteristic zero: if a hypersurface is within the strong monomial case we prove that a resolution can be achieved in a combinatorial manner.

Keywords

positive characteristicresolution of singularitiesdifferential operatorsRees algebras

MSC classification

Secondary: 14E15: Global theory and resolution of singularities
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 8 , August 2013 , pp. 1267 - 1311
Copyright
© The Author(s) 2013 

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