Cohomology classes responsible for Brauer-Manin obstructions, with applications to rational and K3 surfaces

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dc.contributor.advisorVárilly-Alvarado, Anthonyen_US
dc.creatorNakahara, Masahiroen_US
dc.date.accessioned2019-05-17T14:28:29Zen_US
dc.date.available2019-05-17T14:28:29Zen_US
dc.date.created2018-05en_US
dc.date.issued2018-04-18en_US
dc.date.submittedMay 2018en_US
dc.date.updated2019-05-17T14:28:30Zen_US
dc.description.abstractWe 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.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationNakahara, Masahiro. "Cohomology classes responsible for Brauer-Manin obstructions, with applications to rational and K3 surfaces." (2018) Diss., Rice University. <a href="https://hdl.handle.net/1911/105687">https://hdl.handle.net/1911/105687</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/105687en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectRational pointsen_US
dc.subjectBrauer-Manin obstructionen_US
dc.subjectK3 surfacesen_US
dc.subjectdel Pezzo surfacesen_US
dc.titleCohomology classes responsible for Brauer-Manin obstructions, with applications to rational and K3 surfacesen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematicsen_US
thesis.degree.disciplineNatural Sciencesen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.majorAlgebraic geometryen_US
thesis.degree.nameDoctor of Philosophyen_US
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