Math 622: Toric varieties, Grassmannians, and applications Instructor: Frank Sottile Lectures: TuTh 9:35–10:50 Blocker 161 Course webpage: www.math.tamu.edu/~sottile/teaching/12.1/toric.html Grading: Based on class participation and end of term project.  Office Hours : Tuesdays: 11:00–12:00 By appointment

Among the most accessible classes of algebraic varieties are toric varieties and homogeneous spaces, specifically Grassmannians. This is fortunate for these are also among the most commonly encountered outside of algebraic geometry within mathematics and in the applications of algebraic geometry. Toric varieties in particular are currently widely studied in algebraic geometry and its applications.

In this course we will introduce and develop the elementary theory of these two classes of varieties, emphasizing their fundamental combinatorial nature while focusing on concrete examples and explaining some of their applications.

Because of the elementary nature of these varieties, the prerequisite will be graduate algebra, although courses in commutative algebra and algebraic geometry will be helpful.

I feel that in advanced graduate classes, the students get out of the class what they put in. Consequently, I do not assign regular homework to be collected and marked. (But I will mention exercises, which are intended for you to fill in gaps in the presentation, thereby learning a bit more.) At the end of the term, students will present projects; I am starting a file with some suggestions here.

Schedule    Toric Varieties Toric ideals Generation of toric ideals Groebner bases Computation Affine toric varieties Projective toric varieties Orbit decomposition and Limiting behavior Lattice Polytopes Algebraic-Combinatorial Geometric Dictionary Normality of Sturmfellian Toric Varieties Real projective toric varieties Irrational toric varieties Algebraic moment map Geometric combinatorics, lattices and fans Abstract Construction of Toric Varieties Cox/Delzant quotient construction Cox Coordinate Ring Line bundles on toric varieties Toric Degenerations    Grassmannians Plücker embedding equations straightening laws Sagbi degeneration Degree of Grassmannian Schubert decomposition Wronskians Schubert calculus

Text Book:     I do not plan to follow a text book for this class. I will be writing notes throughout the semester, these are for part of a book I am writing with Thorsten Theobald, Applicable Algebraic Geometry. The sections on toric varieties will be written this semester, while the sections on Grassmannians are being revised. These will be made available as they are completed.     While there are several attractive introductions to toric varieties (see below), I am afraid that I cannot recommend a single source for Grassmannians, as there has not been a definitive treatment of these, except as chapters in various books. Other Sources: I can recommend most highly a monumental new book Toric Varieties by Cox, Little, and Schenck. This is an encyclopedia of toric varieties, with a wealth of material, far more than can be digested in a term. This is and will be the definitive source for the subject for many years to come. It is published by the American Mathematical Society Another popular book is Fulton's Introduction to Toric Varieties, by Princeton University Press. Its publication in the 1990's made the subject accessible and really opened up the field. It is not as elementary as the book by Cox, Little, and Schenck or that by Ewald, but it is much shorter and gives a good introduction to the subject. This book by Ewald treats both the algebraic geometry of toric varieties, and the related geometric combinatorics. Many from combinatorics find it the right introduction to the subject.