Arithmetic of affine del Pezzo surfaces

Leiden Repository

Arithmetic of affine del Pezzo surfaces

Type: Doctoral Thesis
Title: Arithmetic of affine del Pezzo surfaces
Author: Lyczak, J.T.
Issue Date: 2019-10-01
Keywords: Algebraic geometry
Arithmetic geometry
Brauer groups
Brauer-Manin obstruction
Del Pezzo surfaces
Integral points
Integral Hasse principle
Log K3 surfaces
Abstract: In this thesis integral points on affine del Pezzo surfaces are studied. The first two chapters offer a review of arithmetic techniques and del Pezzo surfaces, but also a novel approach to del Pezzo surfaces using a new type of surface, namely the peculiar del Pezzo surface. This allows one to study del Pezzo surfaces by considering linear subsystems of cubic plane curves.In Chapter 3 a uniform bound is given for the Brauer group of certain affine del Pezzo surfaces over number fields. This answers an open question for the geometrically related K3 surfaces. While determining this bound techniques are described for (partially) computing these groups.Chapter 4 begins with constructing models of del Pezzo surfaces; not by geometrically manipulating the projective plane over the rationals and taking the flat closure over the integers, but by manipulating schemes over the integers. The advantage of this approach is that one can control the reduction of the surface over all primes. Using these techniques and the computations from Chapter 3 we describe families of surfaces with an order 5 Brauer-Manin obstruction to the integral Hasse principle. These stand out against previously published examples, since these were all of lower order.
Promotor: Supervisor: Stevenhagen P. Co-Supervisor: Bright M.J.
Faculty: Science
University: Leiden

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application/pdf Title Page_Contents 324.8Kb View/Open
application/pdf Chapter 01 474.9Kb View/Open
application/pdf Chapter 02 635.8Kb View/Open
application/pdf Chapter 03 280.9Kb View/Open Full text at publisher site
application/pdf Chapter 04 845.0Kb View/Open
application/pdf Bibliography 179.7Kb View/Open
application/pdf Summary 204.1Kb View/Open
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application/pdf Acknowledgements_CV 117.7Kb View/Open
application/pdf Propositions 108.2Kb View/Open

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