Toric varieties are algebraic varieties defined by combinatorial data, and there is a wonderful interplay between algebra, combinatorics and geometry involved in their study. Many of the key concepts of abstract algebraic geometry (for example, constructing a variety by gluing affine pieces) have very concrete interpretations in the toric case, making toric varieties an ideal tool for introducing students to abstruse concepts.
The first week will cover basic material, including affine toric varieties, projective toric varieties, normal toric varieties constructed from fans, divisors, and homogeneous coordinates. We will also discuss toric surfaces.The second week will go deeper into the subject, covering topics such as ampleness, vanishing theorems in cohomology, the secondary fan, and geometric invariant theory.
An important feature of the workshop is that it does not assume that students have a full background in algebraic geometry. Students will need to know basic facts about varieties in affine and projective space, but we assume no knowledge of schemes, sheaves, cohomology, etc.
Bibliography (PDF 13KB) |
- David Cox, Toric Varieties lecture 1 June 15,2009, 09:30 AM to 10:30 AM
- David Cox, Toric Varieties lecture 2 June 15,2009, 11:00 AM to 12:00 PM
- David Cox, Toric Varieties lecture 3 June 16,2009, 09:30 AM to 10:30 AM
- David Cox, Toric Varieties lecture 4 June 16,2009, 11:00 AM to 12:00 PM
- David Cox, Toric Varieties lecture 5 June 17,2009, 09:30 AM to 10:30 AM
- David Cox, Toric Varieties lecture 6 June 17,2009, 11:00 AM to 12:00 PM
- David Cox, Toric Varieties lecture 7 June 18,2009, 09:30 AM to 10:30 AM
- David Cox, Toric Varieties lecture 8 June 18,2009, 11:00 AM to 12:00 PM
- David Cox, Toric Varieties lecture 9 June 19,2009, 09:30 AM to 10:30 AM
- David Cox, Toric Varieties lecture 10 June 22,2009, 09:30 AM to 10:30 AM
- David Cox, Toric Varieties lecture 11 June 22,2009, 11:00 AM to 12:00 PM
- David Cox, Toric Varieties lecture 12 June 23,2009, 09:30 AM to 10:30 AM
- David Cox, Toric Varieties lecture 13 June 23,2009, 11:00 AM to 12:00 PM
- David Cox, Toric Varieties lecture 14 June 24,2009, 09:30 AM to 10:30 AM
- David Cox, Toric Varieties lecture 15 June 24,2009, 11:00 AM to 12:00 PM
- David Eisenbud, Toric Varieties- D. Eisenbud June 24,2009, 02:00 PM to 03:00 PM
- David Cox, Toric Varieties lecture 16 June 25,2009, 09:30 AM to 10:30 AM
- David Cox, Toric Varieties lecture 17 June 25,2009, 11:00 AM to 12:00 PM
- Matthias Beck, Toric Varieties- M. Beck June 25,2009, 02:00 PM to 03:00 PM
- David Cox, Toric Varieties lecture 18 June 26,2009, 09:30 AM to 10:30 AM
- David Cox, Toric Varieties lecture 19 June 26,2009, 11:00 AM to 12:00 PM
- Sam Payne, Toric Varieties- S. Payne June 26,2009, 02:00 PM to 03:00 PM
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