Toric Geometry Seminar

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We meet on Wednesdays at 4pm in Sloan 159.

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	Additional Resources
Oct. 5	Chris Perez	Affine Toric Varieties
Oct. 12	Zeb Brady	Toric Varieties and Fans
Oct. 19	Jeff Manning	Orbit-Cone Correspondence
Oct. 26	Brian Hwang	Toric Morphisms
Nov. 2	Dori Bejleri	Divisors
Nov. 9	Dori Bejleri	Divisors on Toric Varieties
Nov. 16	Michel van Garrel	Line Bundles
Nov. 23	Michel van Garrel	More on Line Bundles
Nov. 30	Gjergji Zaimi	Polytopes
Dec. 6	Gjergji Zaimi	Polytopes
Jan. 18	Dori Bejleri	Blow-ups, resolution of singularities
Jan. 25	Ryan	Properness, Chow's lemma
Feb. 1	Anton	Cox construction
Feb. 8	Gjergji	Topology of toric varieties
Feb. 22	Gjergji	Moment map
Feb. 29	Brian Hwang	The Symplectic Viewpoint (Toric Manifolds)
Future	?	Tropical Geometry
Future	?	Differential forms, Canonical sheaf

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
• relation between tropical geometry and toric geometry.
• toric geometry and Ehrhart polynomials
• the moment map

• smoothness
• simplicialness
• properness of a map
• fiber bundles
• blow-ups

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?
• When is a toric set an affine toric variety?

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
• Tadao Oda's Convex Bodies and Algebraic Geometry
• Ecole d'Eté 2000 : Géométrie des variétés toriques

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.

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