Configurations of Kodaira Fibers on Elliptic Surfaces

The study of algebraic surfaces has always been a central field in Algebraic Geometry: since the Italian geometers, Castelnuovo and Enriques above all, started at the end of 19th century the impressive theory known today as the Enriques classification of surfaces, they have never stopped asking mathematicians new fascinating questions. Following the modern terminology, we can coarsely divide algebraic surfaces into 4 big classes, depending on the asymptotic behaviour of the plurigenera $P_n$, and identified by the Kodaira dimension $\kappa\in \{-\infty,0,1,2\}$. Although the early geometers had a good understanding of the first two classes (namely, of the surfaces with all $P_n \le 1$), they didn't know much about those with $\kappa=1$, except that they admitted an elliptic fibration, that is, a fibration onto a curve with almost all fibers smooth of genus 1; these surfaces are called elliptic. It wasn't until the sixties, that Kodaira provided an exhaustive study of them: he classified the possible singular fibers and he introduced several fundamental tools, as the monodromy around singular fibers. Nowadays elliptic surfaces are quite well understood, but they are a neverending source of examples: for instance, many rational, K3 and Enriques surfaces admit an elliptic fibration, and many of their features can be understood using this fibration. They play a fundamental role in many geometric and arithmetic problems, and although they do not require too many prerequisites, they allow many applications of distinct fields of mathematics, such as graph theory, lattice theory, and the theory of singularities. In this thesis we deal with Jacobian elliptic surfaces, i.e. elliptic surfaces admitting a section. This makes our treatment much less general, but it allows us to explain all the main geometric features of elliptic surfaces and to present the Mordell-Weil group of sections, from which we derive several fundamental consequences. Moreover, following Kodaira [1], we provide a complete study of germs of singular fibers: we classify them analytically, according to the orders of vanishing of 3 special sections, we compute their monodromy and the root lattice they induce. At the end of the thesis we focus on a problem investigated by Beauville, Miranda and Persson (see [2], [3], [4], [5], [6]): the classification of possible configurations of singular fibers on elliptic surfaces. In other words, given a list of Kodaira fibers, is there an elliptic surface with exactly those singular fibers? We only limit ourselves to the manageable cases, namely the rational and K3 cases, as the others admit an insanely high number of possible configurations; however, most of the techniques we introduce are general, and they could be used to study the problem in much more complex situations