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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Existence of curves with prescribed topological singularities

Authors: Thomas Keilen and Ilya Tyomkin
Journal: Trans. Amer. Math. Soc. 354 (2002), 1837-1860
MSC (2000): Primary 14H10, 14H15, 14H20; Secondary 14J26, 14J27, 14J28, 14J70
Published electronically: December 3, 2001
MathSciNet review: 1881019
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Abstract: Throughout this paper we study the existence of irreducible curves $C$ on smooth projective surfaces $\Sigma$ with singular points of prescribed topological types $\mathcal S_1,\ldots,\mathcal S_r$. There are necessary conditions for the existence of the type $\sum_{i=1}^r \mu(\mathcal S_i)\leq \alpha C^2+\beta C.K+\gamma$ for some fixed divisor $K$on $\Sigma$ and suitable coefficients $\alpha$, $\beta$ and $\gamma$, and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case $\Sigma=\mathbb P_{\mathbb C}^2$. An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up $\Sigma$ of the form $\mathcal O_{\widetilde{\Sigma}}(\pi^*D-\sum_{i=1}^rm_iE_i)$, deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in $\mathbb P_{\mathbb C}^3$, and K3-surfaces.

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Additional Information

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

Ilya Tyomkin
Affiliation: Tel Aviv University, School of Mathematical Sciences, Ramat Aviv, Tel Aviv 69978, Israel

Keywords: Algebraic geometry, singularity theory
Received by editor(s): December 1, 2000
Received by editor(s) in revised form: March 22, 2001
Published electronically: December 3, 2001
Additional Notes: The first author was partially supported by the DFG Schwerpunkt “Globale Methoden in der komplexen Geometrie”. The second author was supported in part by the Herman Minkowsky–Minerva Center for Geometry at Tel–Aviv University, and by grant no. G0419-039.06/95 from the German-Israeli Foundation for Research and Development.
The authors would like to express their thanks to Gert-Martin Greuel, Christoph Lossen, and Eugenii Shustin for bringing the subject to their attention and for many helpful discussions.
Article copyright: © Copyright 2001 American Mathematical Society