The Brauer–Manin obstruction and Sha[2]
M. J. Bright, N. Bruin, E.V. Flynn and A. Logan
Abstract: 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.
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