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The Brauer-Manin obstruction and \(\text{Ш}[2]\). (English) Zbl 1222.11084
Summary: We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We 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). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of \(\text{Ш}[2]\), for families of curves of genus 2 of the form \(y^{2}=f(x)\), where \(f(x)\) is a quintic containing an irreducible cubic factor.

11G35 Varieties over global fields
14F22 Brauer groups of schemes
14J26 Rational and ruled surfaces
Full Text: DOI
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