Intermediate Jacobians and rationality over arbitrary fields

Abstract

We prove that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k.To do so, we develop a theory of intermediate Jacobians for geometrically rational threefolds over arbitrary, not necessarily perfect, fields.As a consequence, we obtain the first examples of smooth projective varieties over a field k which have a k-point, and are rational over a purely inseparable field extension of k, but not over k.representative Ab 2 (X) for algebraically trivial codimension 2 cycles on X (see §1.2 of op.cit.for the definition).However, when k is imperfect, it is not known whether Ab 2 (X) k is isomorphic to Ab 2 (X k ).For this reason, we do not know how to construct on Ab 2 (X) the principal polarization that is so crucial to the Clemens-Griffiths method.To overcome this difficulty and prove Theorem A in full generality, we provide, over an arbitrary field k, an entirely new construction of an intermediate Jacobian.Our point of view is inspired by Grothendieck's definition of the Picard scheme (for which see [FGA], [BLR90, Chapter 8], [Kle05]).With any smooth projective k-rational threefold X over k, we associate a functor CH 2 X/k,fppf : (Sch/k) op → (Ab) endowed with a natural bijection CH 2 (X k ) ∼ -→ CH 2 X/k,fppf (k) (see Definition 2.9 and (3.1)).The functor CH 2 X/k,fppf is an analogue, for codimension 2 cycles, of the Picard functor Pic X/k,fppf .Too naive attempts to define the functor CH 2 X/k,fppf on the category of k-schemes, such as the formula "T → CH 2 (X T )", fail as Chow groups of possibly singular schemes are not even contravariant with respect to arbitrary morphisms: one would need to use a contravariant variant of Chow groups (see Remark 3.2 (ii)).To solve this issue, we view Chow groups of codimension ≤ 2 as subquotients of K-theory by means of the Chern character, and we define CH 2 X/k,fppf as an appropriate subquotient of (the fppf sheafification of) the functor T → K 0 (X T ).That this procedure gives rise to the correct functor, even integrally, is a consequence of the Riemann-Roch theorem without denominators [Jou70].We show that CH 2 X/k,fppf is represented by a smooth k-group scheme CH 2

Locations

• Annales Scientifiques de l École Normale Supérieure - View
• arXiv (Cornell University) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View
• DataCite API - View