The Brauer-Manin obstruction and Sha[2]

The authors of this paper discuss the Brauer–Manin obstruction on del Pezzo surfaces of degree 4. They outline a detailed algorithm for computing the obstruction, and provide associated programs in MAGMA. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). The relationship with the Tate–Shafarevich group is exploited to give new types of examples of Sha[2], for families of curves of genus 2 of the form y2 = f(x), where f(x) is a quintic containing an irreducible cubic factor.