The Clemens–Griffiths method over non-closed fields

Cycle Class Maps and Birational Invariants

“…The Galois action on the Néron-Severi group can still be used to obtain nonrationality in some cases, but never when that group has rank 1 or is split over the ground field. The case of complete intersections of two quadrics was considered in depth in [19]; we gave a complete characterization of rationality over k h R. Benoist and Wittenberg [6] developed an approach inspired by the Clemens-Griffiths method of intermediate Jacobians.…”

Rationality over non-closed fields of Fano threefolds with higher geometric Picard rank

“…In this section we prove non-rationality of Fano threefolds of type X (3,3) . We will use the following reformulation of a result of Benoist-Wittenberg from [BW20].…”

“…We also check that the discriminant double covering Γ → Γ obtained from this conic bundle is trivial over a quadratic extension k ′ of the base field k but nontrivial over k and that the conic bundle has a rational section over k ′ . We check in Theorem 6.10 that these geometric properties characterize the non-rational conic bundles constructed by Benoist and Wittenberg in [BW20] and deduce in Corollary 6.11 non-rationality of X from [BW20, Proposition 3.4].…”

Decomposition of the diagonal, intermediate Jacobians, and universal codimension-2 cycles in positive characteristic

“…The goal of this paper is to consider these types of questions in positive characteristic. The Clemens-Griffiths results on rationality have been investigated in this setting, for instance in [Mur73,Bea77] over algebraically closed fields, and more recently in [BW19a,BW19b] over arbitrary fields. In this paper we focus on the topic of stable rationality, with a focus on Voisin's conditions (5) and (6).…”