We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class β of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C 8 ∩ C 14 has five irreducible components. In the component corresponding to the existence of a tangent conic, we prove that the general member is both pfaffian and has β nontrivial. Such cubic fourfolds provide twisted derived equivalences between K3 surfaces of degrees 2 and 14, hence further corroboration of Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds.

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.Comment: 24 pages, 3 figures, Magma scripts included at the end of the source file

We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behavior of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By importing work of Chambert-Loir and Tschinkel to our setup, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups. Contents

Abstract. We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class α that is unramified at every finite prime, but ramifies at real points of X. Motivated by Hodge theory, the pair (X, α) is constructed from a double cover of P 2 × P 2 ramified over a hypersurface of bi-degree (2, 2).

We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface Y of degree 2 over Q together with a three torsion Brauer class α that is unramified at all primes except for 3, but ramifies at all 3-adic points of Y . Motivated by Hodge theory, the pair (Y, α) is constructed from a cubic fourfold X of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for α. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold X (and hence the fibers) over Q 3 and local solubility at all other primes. 1 Brauer-Manin obstructions to the Hasse principle (see §6 for details):the constant classes Br 0 (X) := im (Br(k) → Br(X)) , and the algebraic classes Br 1 (X) := ker Br(X) → Br(X) . Elements ofBr(X) that are not algebraic are called transcendental. Brauer classes of odd order, whether algebraic or transcendental, had previously only been known to obstruct weak approximation on K3 surfaces [Pre13, IS15]. Ieronymou and Skorobogatov [IS15] showed that elements of odd order can never obstruct the Hasse principle on smooth diagonal quartics in P 3 Q . Their work prompted them to ask if there exists a locally soluble K3 surface over a number field with an odd-order Brauer-Manin obstruction to the Hasse principle [IS15, p. 183]. Since then, Skorobogatov and Zarhin [SZ16] have shown that such classes cannot obstruct the Hasse principle on Kummer surfaces. Recently, Corn and Nakahara [CN17] gave a positive answer to Ieronymou and Skorobogatov's question, by showing that a 3-torsion algebraic class can obstruct the Hasse principle on a degree 2 K3 surface over Q. Our main result shows that 3-torsion transcendental Brauer classes can obstruct the Hasse principle on K3 surfaces. Theorem 1.1. There exists a K3 surface Y over Q of degree 2, together with a class α in Br Y [3] such that Y (A Q ) = ∅ and Y (A Q ) {α} = ∅.Moreover, Pic Y ∼ = Z, and hence Br 1 (Y )/ Br 0 (Y ) = 0. In particular, there is no algebraic Brauer-Manin obstruction to the Hasse principle on Y .