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

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 $V_{irr}(d; d_1,\ldots ,d_r; k)$.