Enumerating curves on rational surfaces: the rational fibration method

TL;DR

The paper introduces a straightforward method for counting rational curves on rational surfaces, providing new proofs and solutions for classical enumerative geometry problems involving plane curves and Hirzebruch surfaces.

Contribution

It presents a novel, simple approach called the rational fibration method for enumerating rational curves on rational surfaces, simplifying existing proofs and solving new problems.

Findings

Short proof of Kontsevich's formula for plane curves

Solution to enumerative problem on Hirzebruch surface F_3

Introduction of the rational fibration method

Abstract

A new, simple method to approach enumerative questions about rational curves on rational surfaces is described. Applications include a short proof of Kontsevich's formula for plane curves and a the solution of the analogous problem for the Hirzebruch surface F_3.