Introduction to Toric Geometry

Toric varieties form a well-understood and intensively studied class of algebraic varieties. They provide a rich source of examples and test cases for theorems and conjectures. Moreover, they have direct applications in physics and in polynomial system solving. For instance, compact, projective toric varieties are the natural generalization of projective space considered in the study of discriminants and resultants for sparse polynomials. The theory consists of a nice interplay between algebra, geometry and combinatorics.

In this course, we will start from embedded affine toric varieties via monomial maps to later discuss standard constructions of toric varieties from cones, fans and polytopes. We will motivate the theory by insights from sparse polynomial system solving, and (time permitting) present more advanced constructions such as the Cox ring and line bundles on toric varieties. Some important theorems and constructions that are featured include the orbit-cone correspondence, the Bernstein-Khovanskii-Kushnirenko theorem and the construction of a toric variety as a GIT (Geometric Invariant Theory) quotient.

Date and time info
Wednesday, 10.00-12.00

Keywords
toric varieties, toric geometry

Prerequisites
Basic algebraic geometry, at the level of introductory text books such as `Ideals, Varieties and Algorithms'.