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.
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 |
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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 |
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Exercise sheet 1 |
Exercise sheet 2 |
Exercise sheet 3 |
Exercise sheet 4 |
Exercise sheet 5 |
Exercise sheet 6 |