Biquadratic addition laws on elliptic curves in $\mathbb{P}^3$ and the canonical map of the $(1,2,2)$-Theta divisor

Abstract

We recall that a smooth ample surface $\mathcal{S}$ in a general $(1,2,2)$-polarized abelian threefold, which is the pullback of the Theta divisor of a smooth plane quartic curve $\mathcal{D}$, is a surface isogenous to the product $\mathcal{C} \times \mathcal{C}$, where $\mathcal{C}$ is a genus $9$ curve embedded in $\mathbb{P}^3$ as complete intersection of a smooth quadric and a smooth quartic. We show that the space of global holomorhic sections of the canonical bundle of this surface is generated by certain determinantal bihomogeneous polynomials of bidegree $(2,2)$ on $\mathbb{P}^3$, which can be used to define biquadratic addition laws on the Jacobi model of elliptic curves, embedded in $\mathbb{P}^3$ as complete intersection of two quadrics. Finally, we use this interesting relationship with the biquadratic addition laws to describe the behavior of the canonical map of $\mathcal{S}$.