Rational curves on the compactified moduli space of rational curves with marked points

TL;DR

This paper studies the homological classes of rational curves on the moduli space of rational nodal curves with marked points, describing polyhedral cones for specific cases and identifying which classes correspond to actual rational curves.

Contribution

It explicitly describes the polyhedral cones of homology classes for $X_5$ and $X_6$, and determines which classes correspond to rational curves.

Findings

Homology classes form polyhedral cones in $X_5$ and $X_6$

Not all classes in the cones are rational curves

Classes far from the cone walls are rational curves

Abstract

We investigate the possible homological classes of rational curves on the moduli space $X_n=\bar{\mathcal{M}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to certain polyhedral cones which we describe explicitly. Not all homology classes in the cone correspond to rational curves, but all classes sufficiently far from the walls of the cone do. The proof for $X_5$ is explicit, the proof for $X_6$ was carried out using computers.