Conic bundle fourfolds with nontrivial unramified Brauer group

Abstract

We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P 3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen-Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P 2 . We also prove the existence of universally CH0-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.

Universal Triviality of the Chow Group of 0-cycles and the Brauer Group

“…The Brauer group, and more generally, unramified cohomology groups, have been used most notably in the context of the Lüroth problem and in Noether's problem, see [A-M72], [Sa77], [Bogo87], [CTO], [CT95], [Bogo05], [Pey08]. The Chow group of 0-cycles has gained increased attention since the advent of the degeneration method due to Voisin [Voi15], and further developed by Colliot-Thélène and Pirutka [CT-P16], which unleased a torrent of breakthrough results proving the non stable rationality of many types of conic bundles over rational bases [HKT15], [A-O16], [ABBP16], [BB16], hypersurfaces of not too large degree in projective space [To16], [Sch18], and many other geometrically interesting classes of rationally connected varieties, e.g., [CT-P16], [HT16], [HPT16]. See [Pey16] for a survey of many of these results.…”

Introduction to work of Hassett-Pirutka-Tschinkel and Schreieder

“…We thus have a parameter space B given by a projective space of dimension 59 (the corresponding vector space being given by the coefficients of 10 quadratic forms in three variables). We have the map X→B whose fibres X m are the various quadric bundles X m →P 2 C , for X m ⊂ P 3 C × P 2 C given by the vanishing of a nonzero complex bihomogeneous form of bidegree (2,2).…”

Stable rationality of quadric and cubic surface bundle fourfolds

“…An integral variety X over a field k is stably rational if X × P m is rational, for some m. In recent years, failure of stable rationality has been established for many classes of smooth rationally connected projective complex varieties, see, for instance [1,3,4,7,15,16,22,23,24,25,26,27,28,29,30,33,34,36,37]. These results were obtained by the specialization method, introduced by C. Voisin [37] and developed in [16].…”

“…This provided a first example showing that rationality is not a deformation invariant for smooth projective complex varieties. A detailed analysis of this same reference variety, from the point of view of the conic bundle structure obtained by projection onto the second factor, is made in [3]. Recently, Schreieder [33] developed a refinement of the specialization method, relaxing the condition that the reference variety admits a universally CH 0 -trivial resolution.…”

“…Theorem 1. The very general hypersurface of bidegree (2,3) in P 2 × P 3 over C is not stably rational.…”