Genus g Gromov-Witten invariants of Del Pezzo surfaces: Counting plane curves with fixed multiple points
Abstract
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.