Geometry of families of nodal curves on rational surfaces

Details Geometry of families of nodal curves on rational surfaces Geometry of families of nodal curves on rational surfaces

1996-01-19

arxiv.org/abs/alg-geom/9601020v1

Mathematics

Algebraic Geometry

AMSLaTeX v 1.2

Scientific paper

Let P^2_r be the projective plane blown up at r generic points. Denote by E_0,E_1,...,E_r the strict transform of a generic straight line on P^2 and the exceptional divisors of the blown-up points on P^2_r respectively. We consider the variety V of all irreducible curves C in |dE_0-d_1E_1-...-d_rE_r| with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptyness. Moreover, we extend our conditions for the smoothness and the irreducibility on families of reducible curves. For r<10 we give the complete answer concerning the existence of nodal curves in V.

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