Geometry and arithmetic of del Pezzo surfaces of degree 1
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.
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. 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.
