Irreducible components of moduli spaces of maps to smooth projective toric varieties in genus 0

TL;DR

This paper provides a combinatorial description of the irreducible components of genus 0 moduli spaces of maps to smooth projective toric varieties, including concrete examples and the first such description for non-projective space targets.

Contribution

It introduces a novel combinatorial approach to describe irreducible components of these moduli spaces, extending understanding beyond projective spaces.

Findings

Describes irreducible components of moduli spaces for toric varieties

Provides concrete example with blow-up of projective plane

First description of smoothable locus for non-projective targets

Abstract

We give a combinatorial description of the irreducible components of the moduli space $\overline{\mathcal{M}}_{0,n}(X,\beta)$ for a smooth projective toric variety $X$. The result is based on the study of the irreducible components of an abelian cone over a smooth Noetherian Artin stack. We give concrete applications of the result including $\overline{\mathcal{M}}_{0,0}(Bl_{pt} \mathbb{P}^2,2\ell)$, where we also describe the main component. This is the first example where the smoothable locus of $\overline{\mathcal{M}}_{g,n}(X,\beta)$ is described for $X$ not a projective space.