In this thesis we study curves. In the first half, we study moduli spaces of curves and Gromov-Witten invariants, certain kinds of curves counts. We employ logarithmic geometry for this. Some major... Show moreIn this thesis we study curves. In the first half, we study moduli spaces of curves and Gromov-Witten invariants, certain kinds of curves counts. We employ logarithmic geometry for this. Some major results include the polynomiality of the double ramification cycle and recursive relations for the log double ramification cycle.In the second half we study rational points on curves, in particular Chabauty's method for finding the rational points and extensions of it. Major results include that the geometric (quadratic) Chabauty method is theoretically stronger than the original (quadratic) Chabauty method, and that local heights for quadratic Chabauty are explicitly computable. Show less
This thesis contains results on the arithmetic and geometry of del Pezzo surfaces of degree 1.In Chapter 1 we give the necessary background, assuming the reader is familiar with algebraic geometry.... Show moreThis thesis contains results on the arithmetic and geometry of del Pezzo surfaces of degree 1.In Chapter 1 we give the necessary background, assuming the reader is familiar with algebraic geometry. In Chapter 2, which is joint work with Julie Desjardins, we give necessary and sufficient conditions for the set of rational points on a del Pezzo surface of degree 1 from a certain family to be dense with respect to the Zariski topology. In Chapter 3 we study the action of the Weyl group on the E8 root system. The 240 roots in E8 are in one-to-one correspondence with the 240 exceptional curves on a del Pezzo surface of degree 1. We define the complete weighted graph where each vertex represents a root, and two vertices are connected by an edge with a weight defined by the dot product. We prove that for a large class of subgraphs of this graph, any two subgraphs from this class are isomorphic if and only if there is a symmetry of the graph that maps one to the other. We also give invariants that determine the isomorphism type of a subgraph. These results reduce computations on the graph significantly.In Chapter 4 we study the configurations of the 240 exceptional curves on a del Pezzo surface of degree 1, using results from Chapter 3. We prove that a point on a del Pezzo surface of degree 1 is contained in at most 16 exceptional curves in characteristic 2, at most 12 exceptional curves in characteristic 3, and at most 10 exceptional curves in all other characteristics. We give examples that show that the upper bounds are sharp in all characteristics, except possibly in characteristic 5.Finally, in Chapter 5 we show that if at least 9 exceptional curves intersect in a point on a del Pezzo surface S of degree 1, the corresponding point on an elliptic surface constructed from S is torsion on its fiber. This is less trivial than some experts thought. We use a list of all possible configurations of at least 9 pairwise intersecting exceptional curves computed in Chapter 3, and with an example from Chapter 4 we show that the analogue statement is false for 6 or fewer exceptional curves. 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
A common theme in the research on rational points on varieties is: investigating under which conditions rational points are dense with respect to a chosen topology. We prove several existence... Show moreA common theme in the research on rational points on varieties is: investigating under which conditions rational points are dense with respect to a chosen topology. We prove several existence results concerning K3 surfaces defined over the rational numbers with a dense set of rational points with respect to the p-adic topology, for a prime number p, and product topologies arising from these Show less
