Vanishing of the Brauer group of a del Pezzo surface of degree 4

Abstract

The Vanishing of the Brauer Group of a del Pezzo Surface of Degree 4 Manar RimanChair of the Supervisory Committee: Associate Professor Bianca Viray Mathematics A del Pezzo surface X of degree 4 over a field k can be thought of as the smooth complete intersection of 2 quadrics in P 4 . Arithmetic geometers are interested in computing the quotient of its Brauer group Br X{ Br 0 X, where Br 0 X " impBr k Ñ Br Xq. Several algorithms have been implemented to compute this quotient; see [2], [30]. In [30], the algorithm relies on a specific arithmetic input related to the solvability of certain quadrics associated to the pencil determined by X. We explicitly construct a del Pezzo surface X of degree 4 over a field k such that H 1 pk, Pic Xq » Z{2Z while Br X{ Br k is trivial. This proves that the algorithm to compute the Brauer group in [30] cannot be generalized in some cases.

Categories and Metaphysics:

“…Similarly, in [Rim20], Riman considered the case where k = Q cyc (the cyclotomic closure of Q) and where V is a certain del Pezzo surface of degree 4 (see loc. cit.).…”