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Transactions of the American Mathematical Society

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Irreducibility of equisingular families of curves

Author: Thomas Keilen
Journal: Trans. Amer. Math. Soc. 355 (2003), 3485-3512
MSC (2000): Primary 14H10, 14H15, 14H20; Secondary 14J26, 14J27, 14J28, 14J70
Published electronically: April 25, 2003
MathSciNet review: 1990160
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Abstract: In 1985 Joe Harris proved the long-standing claim of Severi that equisingular families of plane nodal curves are irreducible whenever they are nonempty. 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 irreducible. Considering different surfaces, including general surfaces in $\mathbb P_{\mathbb C}^3$ and products of curves, we produce a sufficient condition of the type

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

where $\gamma$ is some constant and $X(\mathcal{S}_i)$ some zero-dimensional scheme associated to the singularity type. Our results carry the same asymptotics as the best known results in this direction in the plane case, even though the coefficient is worse. For most of the surfaces considered these are the only known results in that direction.

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Additional Information

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

Keywords: Algebraic geometry, singularity theory
Received by editor(s): August 10, 2001
Received by editor(s) in revised form: February 5, 2002
Published electronically: April 25, 2003
Additional Notes: The author was partially supported by the DFG-Schwerpunkt “Globale Methoden in der komplexen Geometrie”. The author would like to thank the referee for pointing out Example \ref{ex:referee}.
Article copyright: © Copyright 2003 American Mathematical Society

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