Counting Curves of Any Genus on Rational Ruled Surfaces

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 (Theorem 1.1), and the number of such curves which are irreducible (Theorem 1.3). 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 Fn+2, (ii) the Gromov-Witten invariants are deformation-invariant, and (iii) the Gromov-Witten invariants of F0 and F1 are enumerative, Theorem 1.3 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).