Bright, M. J.; Bruin, N.; Flynn, E. V.; Logan, A. The Brauer-Manin obstruction and \(\text{Ш}[2]\). (English) Zbl 1222.11084 LMS J. Comput. Math. 10, 354-377 (2007). 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. Cited in 321 Documents MSC: 11G35 Varieties over global fields 14F22 Brauer groups of schemes 14J26 Rational and ruled surfaces Keywords:Brauer-Manin obstruction on del Pezzo surfaces of degree 4; algorithm; MAGMA; curves of genus 2 PDF BibTeX XML Cite \textit{M. J. Bright} et al., LMS J. Comput. Math. 10, 354--377 (2007; Zbl 1222.11084) Full Text: DOI References: [1] DOI: 10.1007/BFb0078705 · doi:10.1007/BFb0078705 [2] Colliot-Thélène, J. Reine Angew. Math 373 pp 37– (1987) [3] DOI: 10.1017/S0305004106009844 · Zbl 1114.14010 · doi:10.1017/S0305004106009844 [4] DOI: 10.1017/S0305004198003144 · Zbl 0951.14010 · doi:10.1017/S0305004198003144 [5] DOI: 10.1017/S0305004100076106 · Zbl 0804.14018 · doi:10.1017/S0305004100076106 [6] DOI: 10.4064/aa98-3-4 · Zbl 0972.11058 · doi:10.4064/aa98-3-4 [7] DOI: 10.1007/s002220050291 · Zbl 0951.14013 · doi:10.1007/s002220050291 [8] DOI: 10.1007/s002080050156 · Zbl 0889.11021 · doi:10.1007/s002080050156 [9] Manin, Actes du Congrès International des Mathématiciens 1 pp 401– (1970) [10] DOI: 10.1007/BFb0092227 · doi:10.1007/BFb0092227 [11] Kunyavskiĭ, Mém. Soc. Math. France 113 (1989) [12] Grothendieck, Dix exposés sur la cohomologie des schémas pp 46– (1968) [13] Colliot-Thélène, Ann. Fac. Sci. Toulouse Math 1 pp 295– (1992) · Zbl 0787.14012 · doi:10.5802/afst.751 [14] Colliot-Thélène, J. Reine Angew. Math 374 pp 72– (1987) [15] DOI: 10.1016/j.jnt.2005.10.007 · Zbl 1118.14035 · doi:10.1016/j.jnt.2005.10.007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.