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

摘要

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 in general they are still poorly understood. In this note we show that for the theta divisor on the intermediate Jacobian of a cubic threefold, the Tannaka group is exceptional of type E (6). This is the first known exceptional case, and it suggests a surprising connection with the monodromy of the Gauss map.