Toric Geometry Seminar
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Organizational meeting 4pm Thursday, Sept. 29, in Sloan 153.
We meet on Wednesdays at 4pm in Sloan TBA.
About Toric Varieties
A toric variety is a normal variety $X$ with a dense open algebraic torus $T\subseteq X$ so that $T$ acts on $X$ (in a way that extends $T$'s action on itself). People often imagine fixing $T$ (i.e. fixing a dimension), and then considering various ways of "partially compactifying" it to a toric variety. $T$ can be used to "sniff out" properties of $X$. For example, a regular function on $X$ restricts to a regular function on $T$. We (will) understand regular functions on $T$ very well, so a natural question to ask is, "which regular functions on $T$ come from regular functions on $X$?" By asking and answering these types of questions, one can show that toric varieties are determined by a combinatorial wigit, called a fan.
Much of toric geometry is devoted to building the dictionary between the combinatorics of fans (which are easy to think about) and the geometry of the corresponding toric varieties (which, at least initially, are hard to think about). The advantage of having such a dictionary at your command is that it allows you to quickly generate examples and perform calculations. Even if you are thinking about problems on non-toric varieties, working a few toric examples often reveals key insights. This approach is used to crack hard problems in algebraic geometry, but can also be used when learning the basics of the field.
Schedule
Date | Speaker | Topic |
Oct. 5 | Chris Perez | Affine Toric Varieties |
Oct. 12 | Zeb | Toric Varieties and Fans |
The Future | Brian | Morphisms |
The Future | Jeff Manning | Orbit-Cone Correspondence |
The Future | Dori Bejleri | Divisors |
The Future | Michel van Garrel | Line Bundles |
The Future | Gjergji Zaimi | Polytopes |
Possible Topics
- affine toric varieties
- building a fan out of a toric variety
- building a toric variety out of a fan
- toric morphisms
- the orbit-cone correspondence
- Weil divisors, computing the class group
- Cartier divisors, line bundles, and their polytopes
- How to tell if a linear system is big, nef, ample, base-point free, etc?
- the Cox construction
- cohomology
A list of abstracts from a toric geometry seminar at UC Berkeley years ago: [1]
Questions
- Is Chow's lemma for toric varieties clear? That is, given a fan, can it always be made dual to a polytope by subdividing some of the cones?
MathOverflow questions about toric varieties.
References
- Cox, Little, and Schenck: Toric Varieties
- Fulton: Introduction to Toric Varieties
- Mircea Mustaţă's lecture notes on toric varieties
I (Anton) prepared two talks on toric varieties for a student seminar several years ago. At the time, I knew very little about toric varieties, and preparing these talks really laid the foundations for me. The notes were meant for my eyes only, so they may be difficult to make sense of, but if you can make sense of them (I can try to help), they're a nice 4-page summary of the basics.