Birational Methods in Topology and Hyperkähler Geometry
Final Report Abstract
My DFG project focused on some topics in algebraic geometry, which is a branch of pure mathematics. Loosely speaking, algebraic geometry aims to study geometric objects using methods coming from abstract algebra. More precisely, algebraic geometry is the study of algebraic varieties, that is the sets of solutions of polynomial equations. Even though algebraic geometry had its origin more than two thousand years ago with the study of conic sections such as circles and ellipses, we still cannot completely answer many fundamental questions. Fort this reasons it is one of the most active field in mathematics. In my DFG project I studied two distinct problems using the modern methods of birational geometry, a field of algebraic geometry that in the last decades had a great development. In particular, the technical core of my proposal is based on the so called minimal model program, which is a theory concerning the classification of algebraic varieties, especially in high dimension. Its aim is to find special varieties with nice properties that we call minimal models. The first topic of my project belongs partially to the realm of topology, which is the study of the "basic" shapes of geometric objects: the standard example is that topology explains how to distinguish between a sphere and a doughnut. The problem we investigated was to understand whether certain invariants of a variety (its Chern numbers) which depends on its algebraic structure were actually controlled by "simple" topological data (depending only on the underlying topological manifold). We showed that this is true under suitable assumptions using (also) the minimal model program. During the work on this problem we end up proving (in a different paper) a general result on the boundedness of minimal models which is very interesting from the birational geometry point of view. The second topic of my project (which is actually different from the one originally outlined in my proposal) concerned the geometry of the moduli space of curves. Roughly speaking, a moduli space is a variety that parametrizes other varieties which share the same topology. Moduli spaces are of crucial importance in the study of varieties, since they collect many subtle informations about the variation of the algebraic structures. In particular the moduli space of curves is one of the most studied objects in geometry and is also of interest for the physics community. In this context we constructed many new moduli spaces related via the minimal model program to the moduli space of curves. This helps to determine many geometrical properties of the latter one. For both topics the obtained results are collected in papers published or submitted to international mathematical journals and I have been invited to report on them in several seminars all over the world.
Publications
- A note on the fibres of Mori fibre spaces, Eur. J. Math. 4 (2018), no. 3, 859–878
Luca Tasin, G. Codogni, A. Fanelli and R. Svaldi
(See online at https://doi.org/10.1007/s40879-018-0219-z) - On the first steps of the minimal model program for the moduli space of stable pointed curves
Giulio Codogni, Luca Tasin, Filippo Viviani
(See online at https://doi.org/10.48550/arXiv.1808.00231) - Chern numbers of uniruled threefolds
Stefan Schreieder, Luca Tasin
(See online at https://doi.org/10.48550/arXiv.1906.01397) - On some modular contractions of the moduli space of stable pointed curves
Giulio Codogni, Luca Tasin, Filippo Viviani
(See online at https://doi.org/10.2140/ant.2021.15.1245)
