1997/09/04 by Ravi Vakil, Vakil, Ravi
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.alg-geom/9709004
openalex publication_date 1997/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.