The arithmetic and geometry of fibrations in rational and K3 surfaces

Abstract

This thesis consists of 3 chapters. The first chapter deals with introducing the main theory used in the subsequent chapters. Chapter 1 introduces basic concepts in the theory of elliptic surfaces, such as its main definition, the correspondence with elliptic curves over function fields and the classification of distinct fiber types. Furthermore, specific results on rational and K3 elliptic surfaces are presented. In Chapter 2, we study the rank of an elliptic curve E, defined over the function field k(T), which is given by a Weierstrass equation with coefficients of degree at most 2. This is done by studying the induced conic and elliptic fibrations on its Kodaira–Néron model R. In Chapter 3, we study K3 surfaces X with a non-symplectic automorphism σ ∈ Aut(X) of prime order. We classify distinct elliptic fibrations on X with respect to the action of σ on its respective fibers. Each type of elliptic fibrations is related to a linear system on the minimal resolution of the quocient ˜R = X/σ. When the action of σ on the Néron–Severi group of X fixes the fiber class of an elliptic fibration π, this method allows us to determine which Kodaira types are admissible as its reducible fibers. Furthermore, we are able to determine equations for its generic fiber.