Irrationality and monodromy for cubic threefolds

Abstract

Let M denote the moduli space of smooth cubic threefolds. We prove that the monodromy π 1 (M) → Sp(10; Z) of the 3rd cohomology groups on the fibres of the universal family does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.