UC Berkeley

We construct infinitely many Chatelet surfaces, degree 4 del Pezzo surfaces, and Enriques surfaces that are everywhere locally solvable, but have no global rational points. The lack of rational points on these surfaces is explained by an algebraic Brauer-Manin obstruction. The Enriques surfaces arise as quotients of certain K3 surfaces that are ramified double covers of a degree 4 del Pezzo surface with no rational points. We also construct an algebraic family of Chatelet surfaces over an open subscheme of P^1_Q such that exactly 1 Q-fiber has no Q-points. This example is in stark contrast to the philosophy "geometry controls arithmetic".