Composition law and Nodal genus-2 curves in P^2

TL;DR

This paper applies composition laws and moduli space techniques to count specific genus-2 plane curves with fixed complex structures passing through a set number of points, advancing enumerative geometry methods.

Contribution

It introduces a novel approach combining composition laws with moduli space analysis to enumerate genus-2 curves with prescribed properties in the plane.

Findings

Computed the number of genus-2 degree-d plane curves passing through 3d-2 points

Established a new enumeration method using composition laws and moduli spaces

Provided explicit counts for curves with fixed complex structures

Abstract

Recently, there has been great interest in the application of composition laws to problems in enumerative geometry. Using the moduli space of stable maps, we compute the number of irreducible, reduced, nodal, degree-$d$ genus-$2$ plane curves whose normalization has a fixed complex structure and which pass through $3d - 2$ general points in $\Bbb P^2$.