Examples of Abelian Surfaces with Polarization Type (1,3)

Abstract

In the family

$$A\left( {z_0^4 + z_1^4 + z_2^4 + z_3^4} \right) + 2B\left( {z_0^2z_1^2 + z_2^2z_3^2} \right) + 2C\left( {z_0^2z_1^2 + z_2^2z_3^2} \right) + 2D\left( {z_0^2z_1^2 + z_2^2z_3^2} \right) + 4E{z_0}{z_1}{z_2}{z_3} = 0$$

of quartic surfaces in 4 variables invariant under the level (2,2)-Heisenberg Group H _2,2 we study explicitely two subfamilies,

$${\mathcal{F}_{AB}}: = \left\{ {A = B = 0} \right\}{\text{ and }}{\mathcal{F}_{AE}}: = \left\{ {A = E = 0} \right\},$$

and show that

• 1. To every point in F _AE there corresponds an abelian surface which is a product of elliptic curves; it carries a polarization of type (2,2) and (2,6).

• 2. F _AB is a ℙ₁-bundle over an elliptic curve C _AB .

Both 1 and 2 follow by analysing the configuration of lines lying on each element of F _AE and F _AB .

Our main motivation is to study the singular H _2,2 quartic surfaces arising in the study of the moduli space of abelian surfaces of type (1,3)