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.
Categories: math.AG