Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups

Abstract

We show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical for some projection away from a plane g:X⤏P1, i.e., that every Brauer class is obtained by pullback from an element of Brk(P1). As a consequence, we prove that a Brauer class obstructs the existence of a k-rational point if and only if all k-fibers of g fail to be locally solvable, or in other words, if and only if X is covered by curves that each have no adelic points. Using work of Wittenberg, we deduce that for certain quartic del Pezzo surfaces with nontrivial Brauer group the algebraic Brauer–Manin obstruction is sufficient to explain all failures of the Hasse principle, conditional on Schinzel's hypothesis and the finiteness of Tate–Shafarevich groups. The proof of the main theorem is constructive and gives a simple and practical algorithm, distinct from that in [5], for computing all classes in the Brauer group of X (modulo constant algebras).