Please note: this reading group ended in April 2022 and this page is no-longer being actively maintained.

Toric varieties are a special class of algebraic varieties. By definition, a toric variety \( X \) contains a dense open torus \( T = \mathbb{G}_m^n \) such that the usual multiplicative action \( T \times T \to T \) extends to an action of \( T \) on \( X \). This constrains the geometry of \( X \) to the point where it can be studied using combinatorial methods, allowing one to solve problems or compute invariants easily in the toric setting which are otherwise intractable for arbitrary varieties. This makes toric geometry a useful sandbox for testing out new ideas and conjectures in algebraic geometry, as well as being an important area in its own right.

This reading group, followed the classic Introduction to Toric Varieties by William Fulton. An outline of what was covered each week can be found here.

Prerequisites

We assume participants have taken a previous course in algebraic geometry, at the level of C3.4 Algebraic Geometry or equivalent. We also expect participants either have a working knowledge of scheme theory or are concurrently learning the theory of schemes (for instance, by attending the lectures for C2.6 Introduction to Schemes). A very brief overview of the necessary scheme-theoretic background for the first few weeks will be given in the Week 0 session.

Schedule

Meetings took place every Monday at 17:00 in L4, with one exception; the first meeting took place on Friday of Week 0 on Microsoft Teams at 16:00.

Topic Speaker Notes/Slides Exercises
0. Crash course on schemes Gilles Englebert Slides 0 Exercises 0
1. Introduction and toric fans George Cooper Slides 1 Exercises 1
2. Toric polytopes, local properties I Jakub Wiaterek Notes 2 Exercises 2
3. Local properties II, blowups and resolution of singularities Michał Szachniewicz Notes 3
4. The orbit-cone correspondence, topology of toric varieties Andrew Pollock Slides 4
5. Divisors and line bundles Gilles Englebert Notes 5
6. Sheaf cohomology for toric varieties, sheaves of differentials Andrés Ibáñez Núñez Notes 6
7. Chow groups and the intersection ring George Cooper Slides 7
8. Applications to convex geometry Francesco Ballini

Reading Material

Aside from the book by Fulton, the following are useful references: