|
Toric Geometry SemesterSoSe 2022 Lecturer Roberto Gualdi Type of course (Veranstaltungsart) Vorlesung German title Torische Geometrie Contents Toric varieties are algebraic varieties that come equipped with the action of an open dense algebraic torus. Despite their speciality among the class of varieties, they provide a sufficiently flexible pool of examples: most of the ambient varieties usually considered in algebraic geometry, such as affine spaces, projective spaces and products of those, are indeed toric. The general theory of toric varieties, formalized starting from the seventies, strongly relies on convex geometry, and involves objects like polytopes, convex cones, polyhedral complexes and concave functions. This is the reason that has consecrated toric geometry as a privileged actor in the interactions between algebraic geometry and combinatorics. On the one hand, toric varieties offer a fertile testing ground for problems in algebraic geometry, and they have been used to give combinatorial formulas predicting the number of solutions of a system of polynomial equations. On the other hand, they allow to apply powerful algebro-geometric results to problems in convex geometry, and they have led to the solution of McMullen's conjecture by Stanley characterizing the collection of integer numbers appearing as the number of faces of a simplicial polytope. In this course, we will cover the basic theory of toric varieties, by starting from the development of a vocabulary and toolbox in the convex world and then applying it to construct toric varieties and interpret their algebro-geometric properties in terms of the associated combinatorial data. Literature
Recommended previous knowledge Algebraic Geometry I. It is recommended to take simultaneously the course Algebraic Geometry II Time/Date Monday, Thursday 8-10. Exercise class Wednesday 14-16 Location Monday M104, Thursday M101, exercise M102 Registration
BV, MV, MArGeo ECTS 9 |