Theta constants associated to cubic three folds

TL;DR

This paper explores the relationship between cubic threefolds and theta constants by constructing an explicit inverse period map using Prym varieties, advancing understanding of their complex geometric structures.

Contribution

It provides an explicit expression for the inverse of the period map for cubic surfaces in terms of theta constants, linking intermediate Jacobians and Prym varieties.

Findings

Constructed an isomorphism between J(Y) and a Prym variety.

Expressed the inverse period map using theta constants.

Connected cubic threefolds with complex ball embeddings.

Abstract

For a cubic surface X, by considering the intermediate Jacobian J(Y) of the triple covering Y of the 3-dimensional projective space branching along X, Allcock, Carlson and Toledo constructed a period map per from the family of marked cubic surfaces to the four dimensional complex ball embedded in the Siegel upper half space of degree 5. We give an expression of the inverse of per in terms of theta constants by constructing an isomorphism between J(Y) and a Prym variety of a cyclic 6-ple covering of the projective line branching at seven points.