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

Abstract

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 $\phi :S\to {\mathbb{P}}^2$ of degree $6$. Furthermore, if $|\mathcal{L}|$ is base point free, then ${\phi }_{\ast }{\mathcal{O}}_S={\mathcal{O}}_{{\mathbb{P}}^2}\oplus {\Omega }_{{\mathbb{P}}^2}^1\oplus {\Omega }_{{\mathbb{P}}^2}^1\oplus {\mathcal{O}}_{{\mathbb{P}}^2}(-3)$. Using this decomposition, in this paper we construct the graded coordinate ring of $(S,\mathcal{L},\theta )$, where $\theta :G(\mathcal{L})\stackrel{\sim }{\to }H(1,3)$ is a level structure of canonical type. As a corollary we prove that the moduli space of such triples is rational.