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... Show moreIn 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. Show less
This PhD thesis concerns the topic of arithmetic geometry. We address three different questions and each of the questions in some way is about counting how big some set is or can be. We produce... Show moreThis PhD thesis concerns the topic of arithmetic geometry. We address three different questions and each of the questions in some way is about counting how big some set is or can be. We produce heuristics for counting rational points on surfaces given by one diagonal quartic equation. Our results match with experimental data obtained by van Luijk a few years ago. A different result concerns a certain type of conic bundles over low degree hypersurfaces. We count rational points on the base over which the fibre has rational points. We are able to produce asymptotic results where most results in the literature only produce upper bounds. Moreover we investigate the leading constant in this asymptotic formula, matching it up with expected conjectural behaviour that can be found in the literature. Lastly, we study Brauer groups of Kummer surfaces. We give a way to obtain upper bounds for their sizes. Our way is effective (one only needs to use a formula), but the bounds obtained seem not to be sharp. Our method is based on effective versions of Faltings' theorem on finiteness of abelian varieties. Show less
