The contraction morphism between maps and quasimaps to toric varieties

TL;DR

This paper constructs a morphism between the moduli spaces of stable maps and quasimaps to smooth projective toric varieties, proving surjectivity for Fano cases and introducing a degree concept for basepoints.

Contribution

It introduces a new morphism between moduli spaces of stable maps and quasimaps for toric varieties, including a novel degree notion for basepoints and surjectivity results for Fano varieties.

Findings

Constructed a morphism from stable maps to quasimaps moduli spaces.

Proved surjectivity of the morphism when the toric variety is Fano.

Defined the degree of a quasimap at a base-point and showed quasimaps are determined by this degree and regular extension.

Abstract

Given $X$ a smooth projective toric variety, we construct a morphism from a closed substack of the moduli space of stable maps to $X$ to the moduli space of quasimaps to $X$. If $X$ is Fano, we show that this morphism is surjective. The construction relies on the notion of degree of a quasimap at a base-point, which we define. We show that a quasimap is determined by its regular extension and the degree of each of its basepoints.