Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Cycles of singularities appearing in the resolution problem in positive characteristic


Authors: Herwig Hauser and Stefan Perlega
Journal: J. Algebraic Geom. 28 (2019), 391-403
DOI: https://doi.org/10.1090/jag/718
Published electronically: January 4, 2019
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Abstract | References | Additional Information

Abstract: We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity reappears after the last blowup, with just certain exponents of the defining power series shifted upwards. The construction hence yields a cycle. Iterating this cycle leads to an infinite increase of the residual order of the defining power series. This disproves a theorem claimed by Moh about the stability of the residual order under sequences of blowups. It is not a counterexample to the resolution in positive characteristic since larger centers are also permissible and prevent the phenomenon from happening.


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Additional Information

Herwig Hauser
Affiliation: Faculty of Mathematics, University of Vienna, A-1090 Vienna, Austria
Email: herwig.hauser@univie.ac.at

Stefan Perlega
Affiliation: Faculty of Mathematics, University of Vienna, A-1090 Vienna, Austria
Email: stefan.perlega@univie.ac.at

DOI: https://doi.org/10.1090/jag/718
Received by editor(s): August 23, 2017
Received by editor(s) in revised form: October 7, 2017, November 2, 2017, November 21, 2017, December 12, 2017, and January 1, 2018
Published electronically: January 4, 2019
Additional Notes: Supported by project P-25652 of the Austrian Science Fund FWF
Article copyright: © Copyright 2019 University Press, Inc.