On the abelian fivefolds attached to cubic surfaces

TL;DR

This paper studies the algebraic and geometric properties of abelian fivefolds associated with cubic surfaces, computing their Hodge and monodromy groups to understand symmetries and Galois actions in various characteristics.

Contribution

It provides explicit calculations of Hodge groups in characteristic zero and monodromy groups in arbitrary characteristic for abelian fivefolds linked to cubic surfaces, revealing maximal Galois groups in positive characteristic.

Findings

Hodge groups of associated abelian fivefolds are computed in characteristic zero.

Monodromy groups are determined in arbitrary characteristic.

Galois group of the 27 lines is maximal in general positive characteristic.

Abstract

To a family of smooth projective cubic surfaces one can canonically associate a family of abelian fivefolds. In characteristic zero, we calculate the Hodge groups of the abelian varieties which arise in this way. In arbitrary characteristic we calculate the monodromy group of the universal family of abelian varieties, and thus show that the Galois group of the 27 lines on a suitably general cubic surface in positive characteristic is as large as possible.