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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: November 4, 2004
MathSciNet review: 2140446
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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 $D$ on a smooth projective surface $\Sigma$ it thus makes sense to look for conditions which ensure that the family $V_{\vert D\vert}^{irr}\big(\mathcal{S}_1,\ldots,\mathcal{S}_r\big)$ of irreducible curves in the linear system $\vert D\vert _l$ with precisely $r$ singular points of types $\mathcal{S}_1,\ldots,\mathcal{S}_r$ is T-smooth. Considering different surfaces including the projective plane, general surfaces in $\mathbb{P} _{\mathbb{C} }^3$, products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type

\begin{displaymath}\sum\limits_{i=1}^r\gamma_\alpha(\mathcal{S}_i) < \gamma\cdot (D- K_\Sigma)^2, \end{displaymath}

where $\gamma_\alpha$ is some invariant of the singularity type and $\gamma$ 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 $\gamma_\alpha$-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: http://dx.doi.org/10.1090/S0002-9947-04-03588-3
PII: S 0002-9947(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