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Transactions of the American Mathematical Society
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
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
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
Email: shustin@math.tau.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02055-8
PII: S 0002-9947(98)02055-8
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