Toric Geometry

Toric varieties provide a rich class of examples in algebraic geometry that bridge combinatorics and geometry, making them an ideal starting point for exploring the interplay between these fields. We will introduce toric varieties, study their combinatorial and abstract structures, and examine their basic geometry, including Picard groups, cohomology, and singularities.

Course Details

Schedule. Thursdays 16:15–18:00, HG E 41.

Office hours. Wednesday, 15:30–18:00 or by appointment (office HG J 16.2).

Prerequisites. Some background in algebraic geometry is highly desirable. In the absence of an algebraic geometry background, knowledge of at least one of differential, complex, or symplectic geometry is required.

References

• Notes of the course
• J.-P. Brasselet. Introduction to Toric Varieties, Course notes, 2008
• D. A. Cox, J. B. Little, H. K. Schenck. Toric Varieties. American Mathematical Society, 2011
• W. Fulton. Introduction to Toric Varieties. Princeton University Press, 1993

Course log

• 20 Feb 2025. Organization of the seminar. Introduction and motivation.
• 27 Feb 2025. Cones, faces, monoids. Algebraic varieties.
• 6 Mar 2025. Affine toric varieties [interactive example: double cone].
• 13 Mar 2025. From fans to toric varieties.
• 20 Mar 2025. Orbits and their closure.
• 27 Mar 2025. Geometric properties [example of complete, non-polytopal fan, by Sirawit]. From toric varieties to fans.
• 3 Apr 2025. Polytopes. Divisors.
• 10 Apr 2025. Class and Picard groups. Introduction to sheaf cohomology.
• 17 Apr 2025. Class and Picard groups revisited. Line bundles on toric varieties.
• 8 May 2025. Singularities and their resolutions.
• 15 May 2025. Chow groups, characteristic classes, Riemann–Roch and Pick’s formula.
• 22 May 2025. Stanley’s theorem.

Exercises

Sheet 1, Sheet 2, Sheet 3, Sheet 4, Sheet 5, Sheet 6