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

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).