Irrationality of generic cubic threefold via Weil's conjectures

Abstract

An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo p. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of F3. As a spin-off, it is shown that the 5dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over Fq which attains Perret's and Weil's upper bounds.

Irrationality and monodromy for cubic threefolds

“…This note suggests an alternative perspective on the second part, based on ideas of monodromy. There are many proofs of irrationality of (general) cubic threefolds, via Hodge theory, the Weil conjectures, motivic integration, degeneration of Prym varieties etc [Mur73,Col79,Gwe05,MR18,KT19]. The viewpoint presented here does not yield a new proof but seems philosophically related.…”

The Clemens-Griffiths method over non-closed fields

“…Over algebraically closed fields, several techniques have been used to detect that an intermediate Jacobian is not a Jacobian: the geometry of its theta divisor [19], its automorphism group [8], or the zeta function of one of its specializations over a finite field [49]. To give examples of k-rational varieties that are not k-rational, we need a criterion of a more algebraic nature, which can distinguish between Jacobians of curves and their twists.…”