Toric Construction and Chow Ring of Moduli Space of Quasi Maps from ${\mathbb{P}}^1$ with Two Marked Points to ${\mathbb{P}}^1\times {\mathbb{P}}^1$

In this paper, we present explicit toric construction of moduli space of quasi maps from ${\mathbb{P}}^1$ with two marked points to $$\mathbb{P}$^1 \times $\mathbb{P}$^1$, which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincar'e polynomial for some lower degree cases.