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

We give a combinatorial description of the irreducible components of the moduli space ${\bar{\mathcal{M}}}_{0,n}(X,\beta )$ 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 ${\bar{\mathcal{M}}}_{0,0}(B{l}_{pt}{\mathbb{P}}^2,2\mathrm{\ell })$, where we also describe the main component. This is the first example where the smoothable locus of ${\bar{\mathcal{M}}}_{g,n}(X,\beta )$ is described for $X$ not a projective space.