Lectures on toric ideals and their combinatorics
2018 CIMPA Research School
June 25 - 30, 2018, Zacatecas, MEXICO
Lectures at the Commutative Algebra with Applications to Statistics and Coding Theory, June 25 - July 6, 2018, in Zacatecas, Mexico.
A toric variety is a certain algebraic variety modeled on a convex polyhedron. Due to combinatorial tools, toric varieties are very well-studied and they play an important role in commutative algebra, algebraic geometry and combinatorics. Their corresponding ideals (toric ideals) are binomial and prime. These lectures will provide an introduction to toric varieties. We will focus more on examples and methods to generate such varieties. We will introduce the Gröbner fan of ideals as a natural way to produce toric ideals as initial ideals (of an arbitrary ideal) with respect to weight vectors. The basic results on Gröbner degenerations and relations to triangulations of polytopes will be presented. We also introduce several class of binomial ideals associated to graphs, posets and lattices and study their algebraic and homological properties. The main tools to study these objects are Gröbner basis theory, combinatorics and graph theory.
Throughout the lectures we will point out various open problems and conjectures related to binomial ideals and will try some examples and explicit computations in Macaulay2.
Lecture 1: A glimpse of Gröbner theory and polyhedral geometry (25 June, 2018)
Lecture 2: Toric ideals (26 June, 2018)
Lecture 3: Degree of toric varieties and Ehrhart polynomial (27 June, 2018)
Lecture 4: Toric ideals from lattices (28 June, 2018)
Lecture 5: Toric degenerations of varieties (29 June, 2018)