In this thesis we aim to do two things, in the first three sections we develop some arithmetic intersection theory in the style of Gillet-Soul´e. When doing intersection theory one uses Chow’s...Show moreIn this thesis we aim to do two things, in the first three sections we develop some arithmetic intersection theory in the style of Gillet-Soul´e. When doing intersection theory one uses Chow’s moving lemma to move divisors to rational equivalent ones so that they intersect properly. When doing intersection theory over fields the intersection numbers you get this way by taking degrees only depend on the rational equivalence class of a divisor, however in case of Spec Z the degree of a non-zero rational function is non-zero. This is remedied by in addition to the intersection theory over Spec Z, considering an analogous theory on the complex points. Here we consider smooth hermitian line bundles and green currents associated to divisors. For (green) currents there is a ∗-product which satisfies properties analogous to the product in ordinary intersection theory. We have tried to present the results in a way that showcases the similarities and the results we use in arithmetic intersection theory boil down to similar statements holding for both the intersection product and the ∗-product. The other thing we are interested in is heights. In diophantine geometry heights are used to control the number of rational points, they are used for finiteness statements or describing distributions of infinitely many points for example. First we use the arithmetic intersection theory from section 3 to define a global height for arithmetic varieties. Next in section 4 we work with limits of models in the style of Zhang to accomplish a number of things. First by considering p-adic norms the treatment of the finite primes and the infinite prime become more similar. Second by considering limits of models we enlarge the norms and intersection numbers available to us, for example metrics at infinity don’t have to be smooth anymore. We define local heights for each prime p and show that these converge under some assumptions on the line bundles. We can decompose the global height as a sum of local heights, the global height also converges under some assumptions. We also consider metrics associated to an algebraic dynamical system, i.e. we have a surjective morphism f : X → X of a smooth integral projective variety over Q such that f ∗L ∼= L ⊗d for some line bundle L and some d > 0. By a limit argument we obtain a metric on L that is invariant under f ∗ . In section 5 we apply this when X is an abelian variety, f is multiplication by n > 1 and L is a symmetric line bundle. The height obtained from the invariant metric in this case is the Neron-Tate height and we prove some of its elementary properties.Show less
Affine varieties form the building blocks of algebraic geometry. In this thesis we will define the category of affine varieties as a specific category that is anti-equivalent to the category of...Show moreAffine varieties form the building blocks of algebraic geometry. In this thesis we will define the category of affine varieties as a specific category that is anti-equivalent to the category of finitely generated k-algebras that are also an integral domain. Then we will compare the affine varieties obtained this way with the usual definition of affine varieties. We will also see that we can fix the topology on an affine variety using categorical notions. We will give a description of the construction of varieties out of affine varieties. Again we will see that we can view the topology on a variety in categorical terms.Show less
Deze scriptie zal gaan over meetkundige structuren genaamd projectieve vlakken. Projectieve vlakken komen voor in de projectieve meetkunde, en zijn een speciaal (2-dimensionaal) geval van de...Show moreDeze scriptie zal gaan over meetkundige structuren genaamd projectieve vlakken. Projectieve vlakken komen voor in de projectieve meetkunde, en zijn een speciaal (2-dimensionaal) geval van de objecten, genaamd projectieve ruimtes, die daar bestudeerd worden. Het bijzondere van projectieve vlakken is dat enkele eigenschappen die gelden in alle hoger-dimensionale projectieve ruimtes, zoals de Desargueseigenschap uit de stelling van Desargues, niet altijd meer gelden in de 2-dimensionale gevallen. Ondertussen hebben deze vlakken niet zo’n simpele structuur als 0- of 1-dimensionale projectieve ruimtes. Dit maakt het zeer interessant om projectieve vlakken te bestuderen. Het blijkt dat ieder projectief vlak een achterliggende algebra¨ısche structuur, een zogeheten vlakke ternaire ring, heeft, die veel zegt over bepaalde eigenschappen van het projectieve vlak; bijvoorbeeld of het vlak aan de Desargueseigenschap voldoet. In deze scriptie zullen we deze vlakke ternaire ringen bekijken en afleiden welke eigenschappen van projectieve vlakken corresponderen met de eigenschappen van de bijbehorende vlakke ternaire ringen.Show less
