Low-dimensional singularities with free divisors as discriminants

Buchweitz, Ragnar-Olaf; Ebeling, Wolfgang; Graf von Bothmer, Hans-Christian

doi:10.1090/S1056-3911-08-00508-0

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
MR Author ID: 42840
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
MR Author ID: 724323
Email: bothmer@math.uni-hannover.de, bothmer@uni-math.gwdg.de

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.