Toric Geometry Autumn 2023

Dr. Samouil Molcho
Beat Zurbuchen
Time and Place
Tuesday, 12 - 14, HG G 26.1
Office hours
Samouil Molcho's office hours: Fr, 14 - 15:50, HG F27.6
Beat Zurbuchen's office hours: Mo, 14 - 16, HG J 16.4


Toric Varieties are algebraic varieties with exceptionally many symmetries. As a result, it is pos- sible to understand them completely in terms of combinatorial data, called fans. Consequently, several calculations which are in general intractable for arbitrary algebraic varieties reduce to con- crete, beautiful, and solvable combinatorial problems for toric varieties. In this seminar, we will set up the dictionary between toric varieties and combinatorics, and study the main structures present on an algebraic variety through the perpsective of both algebraic geometry and combina- torics. We will finish with some more specialized topics.

Student responsibilites

Each student is expected to lead two half-sessions (or more, depending on interest and availability), covering a topic from above. Part of the requirement is a meeting with me before the session to go over it, during office hours or by appointment.

Each student is expected to take notes for one session.

Date Content Notes
19. September Introduction - Toric Varieties from Algebraic Geometry and Combinatorics
26. September Convex Geometry, geometric realization of fans
3. October One parameter subgroups, Orbit-Cone Correspondence; Equivalence of definitions Notes, Notes
10. October Morphisms; Proper morphisms, birational morphisms, subdivisions Complement on the valuative criterion, Notes
17. October Singularities and smothness Complement on non-singular varieties, Notes, Notes
24. October Resolution of singularities Notes
31. October Divisors and line bundles on toric varieties Divisors, Picard groups
7. November Cohomology Sheaves and cohomology
14. November Cohomology of line bundles Cohomology of the line bundle, Čech cohomology
21. November Projective toric varieties Projective embeddings of toric varieties, Notes 2
28. November Differential forms on toric varieties, Serre duality Differential forms, Serre duality
5. December Chow groups and Chow rings of toric varieties Chow groups, Chow rings
12. December The Riemann-Roch Theorem Pick's formula, Chern classes
19. December


The exercises are posted here each week.

exercise sheet
Exercise sheet 1
Exercise sheet 2
Exercise sheet 3
Exercise sheet 4
Exercise sheet 5
Exercise sheet 6


  1. G. Kempf , F. Knudsen , D. Mumford , B. Saint-Donat: "Toroidal Embeddings", Lecture Notes in Mathematics, Volume 339, Springer, 1973, Link
  2. Fulton, William: "Introduction to Toric Varieties", (AM-131), Princeton University Press, 1993
  3. Hausen, J. David A. Cox, John B. Little, Henry K. Schenck: “Toric Varieties”, Graduate Studies in Mathematics, Volume 124, Springer, 2011
  4. V. I. Danilov: "The Geometry of Toric Varieties", Russian Math. Surveys, 1978, Link
  5. T. Oda: "Convex Bodies and Algebraic Geometry", MATHE3, Springer, 1988, Link
  6. J.-P. Brasselet: "Introduction to Toric Variteies", Course notes, 2006, Link