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

DOI:
https://doi.org/10.1090/S0002-9947-01-02877-X

Published electronically:
December 3, 2001

MathSciNet review:
1881019

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Abstract | References | Similar Articles | Additional Information

Abstract: Throughout this paper we study the existence of irreducible curves on smooth projective surfaces with singular points of prescribed topological types . There are *necessary* conditions for the existence of the type for some fixed divisor on and suitable coefficients , and , 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 . An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up of the form , 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 , and K3-surfaces.

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Additional Information

**Thomas Keilen**

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

Email:
keilen@mathematik.uni-kl.de

**Ilya Tyomkin**

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

Email:
tyomkin@math.tau.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-01-02877-X

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