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

Ravi Vakil
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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.

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BibTeXRIS

Ravi Vakil. 1997-09-04. Genus g Gromov-Witten invariants of Del Pezzo surfaces: Counting plane curves with fixed multiple points. https://arxiv.org/abs/alg-geom/9709004

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