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

## TL;DR

This paper explores the structure of a special surface in a polarized abelian threefold, showing how its canonical sections relate to biquadratic addition laws on elliptic curves embedded in projective space.

## Contribution

It demonstrates that the canonical bundle's global sections are generated by specific determinantal polynomials, linking the geometry of the surface to addition laws on elliptic curves.

## Key findings

- Canonical sections generated by determinantal bihomogeneous polynomials.
- Biquadratic addition laws define elliptic curve group law in projective space.
- Description of the canonical map's behavior for the surface.

## 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 $\mathcal{S}$.