%0 Journal Article
%T Genus g Gromov-Witten invariants of Del Pezzo surfaces: Counting plane curves with fixed multiple points
%A Ravi Vakil
%J Mathematics
%D 1997
%I arXiv
%X As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of general points in the plane. As a special case, we count the number of irreducible genus $g$ curves in any divisor class $D$ on the blow-up of the plane at up to five points (no three collinear). We then show that these numbers give the genus $g$ Gromov-Witten invariants of the surface. Finally, we suggest a direction from which the remaining del Pezzo surfaces can be approached, and give a conjectural algorithm to compute the genus g Gromov-Witten invariants of the cubic surface.
%U http://arxiv.org/abs/alg-geom/9709004v1