Birational Geometry Of Genus One Fibrations And Stability Of Pencils Of Plane Curves

In the first part of this thesis we give a complete classification of relative log canonical models for genus one fibrations in dimensions two and three. More concretely, we generalize the work in [2] by considering both (i) the case where it is not assumed the existence of a section, but of a multisection instead; and (ii) the case of threefolds in one dimension higher. In the second part, we investigate the stability of pencils of plane curves in the sense of geometric invariant theory. One of our main results relates the stability of a pencil of plane curves P to the log canonical threshold of pairs (P^2,C_d), where C_d is a curve in P, thus extending an idea of Hacking [23] and Kim-Lee [27]. Part of our approach consists in observing that we can sometimes determine whether a pencil P is (semi)stable or not by looking at the stability of the curves lying on it. As a beautiful application, we completely describe the stability of Halphen pencils of index two -- classical geometric objects first introduced by Halphen in 1882 [24]. Inspired by the work of Miranda in [40], we provide explicit stability criteria in terms of the geometry of their associated rational elliptic surfaces.