On the Graded Equations of (1,3)-Abelian Surfaces

Let $S$ be an abelian surface over an algebraically closed field $k$ with characteristic different from $2$ and $3$, and $\mathcal{L}$ a symmetric ample line bundle defining a polarisation of type $(1,3)$. Then the linear system $|\mathcal{L}|$ defines a covering map $\varphi\colon S\rightarrow \mathbb{P}^2$ of degree $6$. Furthermore, if $|\mathcal{L}|$ is base point free, then $\varphi_*\mathcal{O}_S = \mathcal{O}_{\mathbb{P}^2} \oplus \Omega^1_{\mathbb{P}^2} \oplus \Omega^1_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(-3)$. Using this decomposition, in this paper we construct the graded coordinate ring of $(S,\mathcal{L},\theta)$, where $\theta\colon G(\mathcal{L})\xrightarrow{\sim} H(1,3)$ is a level structure of canonical type. As a corollary we prove that the moduli space of such triples is rational.