The Vanishing of the Brauer Group of a del Pezzo Surface of Degree 4

Abstract

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 is the image of Br k in Br X. Several algorithms have been implemented to compute this quotient. The algorithms rely 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(k,Pic X) is isomorphic to ZZ/2 ZZ while Br X/Br k is trivial. This proves that the algorithm to compute the Brauer group cannot be generalized in some cases.