Smoothness of equisingular families of curves

Author:
Thomas Keilen

Journal:
Trans. Amer. Math. Soc. **357** (2005), 2467-2481

MSC (2000):
Primary 14H10, 14H15, 14H20; Secondary 14J26, 14J27, 14J28, 14J70

DOI:
https://doi.org/10.1090/S0002-9947-04-03588-3

Published electronically:
November 4, 2004

MathSciNet review:
2140446

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Abstract | References | Similar Articles | Additional Information

Abstract: Francesco Severi (1921) showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor on a smooth projective surface it thus makes sense to look for conditions which ensure that the family of irreducible curves in the linear system with precisely singular points of types is T-smooth. Considering different surfaces including the projective plane, general surfaces in , products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type

where is some invariant of the singularity type and is some constant. This generalises the results of Greuel, Lossen, and Shustin (2001) for the plane case, combining their methods and the method of Bogomolov instability. For many singularity types the -invariant leads to essentially better conditions than the invariants used by Greuel, Lossen, and Shustin (1997), and for most classes of geometrically ruled surfaces our results are the first known for T-smoothness at all.

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Additional Information

**Thomas Keilen**

Affiliation:
Fachbereich Mathematik, Universität Kaiserslautern, Erwin-Schrödinger-Straße, D – 67663 Kaiserslautern, Germany

Email:
keilen@mathematik.uni-kl.de

DOI:
https://doi.org/10.1090/S0002-9947-04-03588-3

Keywords:
Algebraic geometry,
singularity theory

Received by editor(s):
September 1, 2003

Received by editor(s) in revised form:
December 12, 2003

Published electronically:
November 4, 2004

Additional Notes:
The author was partially supported by the German Israeli Foundation for Research and Development, by the Hermann Minkowski – Minerva Center for Geometry at Tel Aviv University and by EAGER

Article copyright:
© Copyright 2004
American Mathematical Society