Counting curves of any genus on rational ruled surfaces

Abstract

In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of such curves which are irreducible. These numbers are known as Severi degrees; they are the degrees of unions of components of the Hilbert scheme. As (i) $\fn$ can be deformed to $\eff_{n+2}$, (ii) the Gromov-Witten invariants are deformation-invariant, and (iii) the Gromov-Witten invariants of $\eff_0$ and $\eff_1$ are enumerative, Theorem \ref{irecursion} computes the genus $g$ Gromov-Witten invariants of all $\fn$. (The genus 0 case is well-known.) The arguments are given in sufficient generality to also count plane curves in the style of L. Caporaso and J. Harris and to lay the groundwork for computing higher genus Gromov-Witten invariants of blow-ups of the plane at up to five points (in a future paper).

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References in corpus (1)

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

Cited by in corpus (7)

• A conjectural generating function for numbers of curves on surfaces
• Enumerative geometry of hyperelliptic plane curves
• The Symplectic Sum Formula for Gromov-Witten Invariants
• Gromov-Witten Invariants of Symplectic Sums
• Frobenius Manifolds And Virasoro Constraints
• Genus g Gromov-Witten invariants of Del Pezzo surfaces: Counting plane curves with fixed multiple points
• Descendant invariants and characteristic numbers