Geometry Seminar

Date: April 27, 2018

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Jen Berg, Rice

Title: Odd order transcendental obstructions to the Hasse principle on general K3 surfaces

Abstract: Varieties that fail to have rational points despite having local points for each prime are said to fail the Hasse principle. A systematic tool accounting for these failures is called the Brauer-Manin obstruction, which uses [subsets of] the Brauer group, Br X, to preclude the existence of rational points on a variety X. After fixing numerical invariants such as dimension, it is natural to ask which birational classes of varieties fail the Hasse principle, and moreover whether the Brauer group (or certain distinguished subsets) explains this failure. In this talk, we will focus on K3 surfaces, which are relatively simple surfaces in terms of geometric complexity, but whose arithmetic is more mysterious. For example, in 2014 it was asked whether any odd torsion in the Brauer group of a K3 surface could obstruct the Hasse principle. We answer this question in the affirmative; we exhibit a general degree 2 K3 surface Y over the rationals in which an order 3 transcendental Brauer class A obstructs. Motivated by Hodge theory, the pair (Y,A) is constructed from a special cubic fourfold X which admits 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 A. Instead, we prove that a sufficient condition for such a Brauer class to obstruct is insolubility of the fourfold X at 3 and local solubility at all other primes. This is joint work with Tony Varilly-Alvarado.