The arithmetic and geometry of fibrations in rational and K3 surfaces
Surfaces are geometric objects with 2 dimensions (for example, planes, spheres and cylinders). One approach to the study surfaces is to analyze their fibrations, that is, decompositions of the surface into a family of infinitely many curves. When a surface has a fibration, we can use the mathematical theory of curves to determine its arithmetic and geometric properties. In his thesis, Felipe Zingali Meira focuses on two important classes of surfaces: rational surfaces and K3 surfaces.
Rational surfaces can be seen as surfaces which are in some sense similar to planes. Zingali Meira studied rational surfaces which admit two distinct fibrations at the same time: one in conic curves, and one in elliptic curves. Zingali Meira was able to use the interaction between both fibrations to determine an important number related to the elliptic fibration; namely, its Mordell-Weil rank.
K3 surfaces are harder to describe: they can be seen as a generalization of elliptic curves. Zingali Meira studied K3 surfaces which can be transformed into a rational surface by a process in which each points in it is identified with p-1 other points (where p is a prime number). Zingali Meira studied how fibrations in elliptic curves on these K3 surface are related to the fibrations on the rational surface obtained.