Cubic threefolds, Fano surfaces and the monodromy of the Gauss map

TL;DR

This paper reveals that the Tannaka group associated with the theta divisor of the intermediate Jacobian of a cubic threefold is an exceptional E_6 group, linking algebraic geometry with monodromy and exceptional groups.

Contribution

It identifies the Tannaka group for the theta divisor of a cubic threefold's intermediate Jacobian as the first known exceptional E_6 group, highlighting a new connection with Gauss map monodromy.

Findings

Tannaka group for the theta divisor is E_6

First known case of an exceptional Tannaka group

Suggests a link between monodromy of the Gauss map and algebraic groups

Abstract

The Tannakian formalism allows to attach to any subvariety of an abelian variety an algebraic group in a natural way. The arising groups are closely related to moduli questions such as the Schottky problem, but their geometric interpretation is still mysterious. We show that for the theta divisor on the intermediate Jacobian of a cubic threefold, the Tannaka group is an exceptional group of type E_6. This is the first known exceptional case, and it suggests a connection with the monodromy of the Gauss map.