Introduction to Toric Geometry


In the fall semester of 2021 (October 2021- February 2022), I taught an introductory course on toric geometry at MPI MiS Leipzig. This course was held partly in person and partly online, and was aimed at advanced master and PhD students. The required background is basic algebraic geometry, at the level of

D. Cox, J. Little and D. O'Shea, Ideals, varieties and algorithms,  corrected fourth edition, Springer, New York, 2018

or other introductory texts. The lecture notes are available here

Short description

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. 


We closely follow the first six chapters of the text book 

David A Cox, John B Little, and Henry K Schenck. Toric varieties. American Mathematical Soc., 2011.

Other references for further reading include

William Fulton. Introduction to toric varieties. Number 131. Princeton University Press, 1993,

Lecture notes by David Cox, `Lectures on toric varieties',

Lecture notes by Frank Sottile, `Ibadan lectures on toric varieties'.

Past lectures

Handwritten notes and Jupyter notebooks can be found here

Lecture 1 (Oct. 13): Motivation and first examples. Affine toric varieties. 

Lecture 2 (Oct. 20): Monomial maps, toric ideals and affine semigroups. 

Lecture 3 (Oct. 27): Affine toric varieties from cones. Points on affine toric varieties and fixed points of the torus action. 

Lecture 4 (Nov. 3): Normality, smoothness and toric morphisms.

Lecture 5 (Nov. 10): Projective toric varieties from monomial maps and toric ideals. Kushnirenko's theorem.

Lecture 6 (Nov. 24): Affine covers and the projective toric variety of a polytope. Normality and smoothness. Normal fans.   

Lecture 7 (Dec. 1): The glueing construction: normal toric varieties from fans.

Lecture 8 (Dec. 8): Limits of one-parameter subgroups and the orbit-cone correspondence.

Lecture 9 (Dec. 15): Toric morphisms: sublattices of finite index, torus factors and blow-ups. 

Lecture 10 (Jan. 5): Background on divisors. Weil divisors on normal toric varieties and computation of the class group. 

Lecture 11 (Jan. 12): Cartier divisors on normal toric varieties and divisoral sheaves. 

Lecture 12 (Jan. 26): Normal toric varieties as GIT quotients. The Cox ring of a normal toric variety.

Lecture 13 (Feb. 2): Homogenization and system solving. The BKK theorem.