Enumerative geometry of hyperelliptic plane curves

TL;DR

This paper develops recursive methods to compute Gromov-Witten invariants of the Hilbert scheme of two points in the plane, enabling enumeration of hyperelliptic plane curves passing through general points.

Contribution

It introduces a recursive approach to compute invariants and applies them to count hyperelliptic curves of given degree and genus.

Findings

Successfully computes Gromov-Witten invariants for the Hilbert scheme of two points.

Provides enumeration formulas for hyperelliptic plane curves.

Demonstrates the effectiveness of virtual class computations in enumerative geometry.

Abstract

We recursively compute the Gromov-Witten invariants of the Hilbert scheme of two points in the plane. By studying the space of stable maps and computing virtual contributions, we use these invariants to enumerate hyperelliptic plane curves of degree d and genus g passing through 3d+1 general points.