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

Authors:	Gert-Martin Greuel Christoph Lossen Eugenii Shustin

Institution:	Universität Kaiserslautern, Fachbereich Mathematik, Erwin-Schrödinger-Straße, D -- 67663 Kaiserslautern, Germany ; Universität Kaiserslautern, Fachbereich Mathematik, Erwin-Schrödinger-Straße, D -- 67663 Kaiserslautern, German ; Tel Aviv University, School of Mathematical Sciences, Ramat Aviv, Tel Aviv 69978, Israel

Abstract:	Let $\mathbb P^2_r\,$ be the projective plane blown up at 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 in $\|dE_0-\sum _{i=1}^{r} d_iE_i\|$ with 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 $V_{irr}(d;\,d_1,\ldots,d_r;\,k)$ .

Keywords:	Nodal curves nodes families of curves blown--up projective space existence equisingular deformation

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