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On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for)

Author: Herwig Hauser
Journal: Bull. Amer. Math. Soc. 47 (2010), 1-30
MSC (2010): Primary 14B05, 14E15, 12D10
Published electronically: October 28, 2009
MathSciNet review: 2566444
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Abstract: Assume that, in the near future, someone can prove resolution of singularities in arbitrary characteristic and dimension. Then one may want to know why the case of positive characteristic is so much harder than the classical characteristic zero case. Our intention here is to provide this piece of information for people who are not necessarily working in the field. A singularity of an algebraic variety in positive characteristic is called wild if the resolution invariant from characteristic zero, defined suitably without reference to hypersurfaces of maximal contact, increases under blowup when passing to the transformed singularity at a selected point of the exceptional divisor (a so called kangaroo point). This phenomenon represents one of the main obstructions for the still unsolved problem of resolution in positive characteristic. In the present article, we will try to understand it.

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Additional Information

Herwig Hauser
Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090, Wien, Austria

Received by editor(s): November 17, 2008
Received by editor(s) in revised form: December 27, 2008, February 12, 2009, and July 10, 2009
Published electronically: October 28, 2009
Additional Notes: Supported within the project P-18992 of the Austrian Science Fund FWF. The author thanks the members of the Clay Institute for Mathematics at Cambridge and the Research Institute for Mathematical Science at Kyoto for their kind hospitality.
Article copyright: © Copyright 2009 American Mathematical Society

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