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 $\mathrm{\mid }\mathcal{L}\mathrm{\mid }$ defines a covering map $\varphi\colon S\rightarrow $\mathbb{P}$^2$ofdegree$6$\mathrm{.}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.