Irreducibility of equisingular families of curves
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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 $V_{|D|}^{irr}\big (\mathcal {S}_1,\ldots ,\mathcal {S}_r\big )$ of irreducible curves in the linear system $|D|_l$ 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{equation*} \sum \limits _{i=1}^r\deg \big (X(\mathcal {S}_i)\big )^2 < \gamma \cdot (D- K_\Sigma )^2, \end{equation*} 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.References
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Additional Information
- Thomas Keilen
- Affiliation: Universität Kaiserslautern, Fachbereich Mathematik, Erwin-Schrödinger-Straße, D–67663 Kaiserslautern, Germany
- MR Author ID: 689521
- Email: keilen@mathematik.uni-kl.de
- 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 2.5.
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 3485-3512
- MSC (2000): Primary 14H10, 14H15, 14H20; Secondary 14J26, 14J27, 14J28, 14J70
- DOI: https://doi.org/10.1090/S0002-9947-03-03304-X
- MathSciNet review: 1990160