Cohomology classes responsible for Brauer-Manin obstructions, with applications to rational and K3 surfaces
Doctor of Philosophy
We study the classes in the Brauer group of varieties that never obstruct the Hasse principle. We prove that for a variety with a genus 1 fibration, if the generic fiber has a zero-cycle of degree d over the generic point, then the Brauer classes whose orders are prime to d do not play a role in the Brauer–Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties. We also analyze the Brauer–Manin obstruction to rational points on the K3 surfaces over Q given by double covers of P^2 ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer–Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.
Rational points; Brauer-Manin obstruction; K3 surfaces; del Pezzo surfaces