Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 

 

Low-dimensional singularities with free divisors as discriminants


Authors: Ragnar-Olaf Buchweitz, Wolfgang Ebeling and Hans-Christian Graf von Bothmer
Journal: J. Algebraic Geom. 18 (2009), 371-406
DOI: https://doi.org/10.1090/S1056-3911-08-00508-0
Published electronically: July 1, 2008
MathSciNet review: 2475818
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Abstract | References | Additional Information

Abstract: We present versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes a free divisor. These families include finite flat maps, versal deformations of reduced curve singularities, and versal deformations of Gorenstein surface singularities in $ \mathbb{C}^5$. It is shown that such free divisors often admit a ``fast normalization'', obtained by a single application of the Grauert-Remmert normalization algorithm. For a particular Gorenstein surface singularity in $ \mathbb{C}^5$, namely the simple elliptic singularity of type $ \widetilde A_{4}$, we exhibit an explicit discriminant matrix and show that the slice of the discriminant for a fixed $ j$-invariant is the cone over the dual variety of an elliptic curve.


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Additional Information

Ragnar-Olaf Buchweitz
Affiliation: Department of Computer and Mathematical Sciences, University of Toronto at Scarborough, Toronto, Ontario M1A 1C4, Canada
Email: ragnar@math.utoronto.ca

Wolfgang Ebeling
Affiliation: Institut für Algebraische Geometrie, Leibniz Universität Hannover, Postfach 6009, D-30060 Hannover, Germany
Email: ebeling@math.uni-hannover.de

Hans-Christian Graf von Bothmer
Affiliation: Institut für Algebraische Geometrie, Leibniz Universität Hannover, Postfach 6009, D-30060 Hannover, Germany
Address at time of publication: Mathematisches Institut, Georg-August-Universität Göttingen, Bunsen str. 3–5, D-37073 Göttingen, Germany
Email: bothmer@math.uni-hannover.de, bothmer@uni-math.gwdg.de

DOI: https://doi.org/10.1090/S1056-3911-08-00508-0
Received by editor(s): February 14, 2007
Received by editor(s) in revised form: September 6, 2007
Published electronically: July 1, 2008
Additional Notes: The authors were partly supported by the DFG Schwerpunkt “Global Methods in Complex Geometry”. The first author was also partly supported by NSERC grant 3-642-114-80 and wishes to thank his alma mater, the University of Hannover, as well as S.-O. Buchweitz-Klingsöhr and D. Klingsöhr for their hospitality during the preparation of this work.

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
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Online ISSN 1534-7486; Print ISSN 1056-3911
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