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Transactions of the American Mathematical Society

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

 
 

 

Geometry of families of nodal curves on the blown-up projective plane


Authors: Gert-Martin Greuel, Christoph Lossen and Eugenii Shustin
Journal: Trans. Amer. Math. Soc. 350 (1998), 251-274
MSC (1991): Primary 14H10
DOI: https://doi.org/10.1090/S0002-9947-98-02055-8
MathSciNet review: 1443875
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Abstract: Let $\mathbb P^2_r$ be the projective plane blown up at $r$ generic points. Denote by $E_0,E_1,\ldots ,E_r$ the strict transform of a generic straight line on $\mathbb P^2$ and the exceptional divisors of the blown–up points on $\mathbb P^2_r$ respectively. We consider the variety $V_{irr}(d; d_1,\ldots ,d_r; k)$ of all irreducible curves $C$ in $|dE_0-\sum _{i=1}^{r} d_iE_i|$ with $k$ nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non–emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility to families of reducible curves. For $r\leq 9$ we give the complete answer concerning the existence of nodal curves in $V_{irr}(d; d_1,\ldots ,d_r; k)$.


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Additional Information

Gert-Martin Greuel
Affiliation: Universität Kaiserslautern, Fachbereich Mathematik, Erwin-Schrödinger-Straße, D – 67663 Kaiserslautern, Germany
MR Author ID: 76830
Email: greuel@mathematik.uni-kl.de

Christoph Lossen
Affiliation: Universität Kaiserslautern, Fachbereich Mathematik, Erwin-Schrödinger-Straße, D – 67663 Kaiserslautern, German
Email: lossen@mathematik.uni-kl.de

Eugenii Shustin
Affiliation: Tel Aviv University, School of Mathematical Sciences, Ramat Aviv, Tel Aviv 69978, Israel
MR Author ID: 193452
Email: shustin@math.tau.ac.il

Keywords: Nodal curves, nodes, families of curves, blown–up projective space, existence, equisingular deformation
Received by editor(s): February 1, 1996
Article copyright: © Copyright 1998 American Mathematical Society