## Final Projects.

For a final project you should choose a topic about toric varieties and either give a 25 minute talk in class or write up a report about it.

Below is a list of topics to choose from, but you can also find a different topic that interests you. This list will be updated with new topics as the semester progresses.

If you are giving a talk, try not to cram too much material into it. In 25 minutes you have barely enough time to explain the definitions and some examples. Your goal should be to explain a notion in toric geometry to other students, not really prove theorems.

The topics below are very general. Each topic has enough material for several talks.

1.  Normal polytopes. A good reference here is the Oberwolfach workshop report. It describes the main problem and many approaches to it. Each of these approaches could be one talk. Here are a couple of topics for talks:
• Give the main definitions, examples and counterexamples. The main conjecture is that all ample line bundles on a smooth toric varieties are projectively normal. This is known in dimension 2, for polytopes that have unimodular covers, for sufficiently large polytopes. It fails for singular toric varieties in general. A generalization of projective normality is the condition N_p (projective normality being property N_0).
• The strongest known result is that a multiple of a line bundle is projectively normal (and an even higher multiple satisfies property N_p). This is discussed in the article by Milena Hering in the report above. The longer article is in the arxiv.
2. Reflexive polytopes. Reflexive polytopes are lattice polytopes whose polar dual is also a lattice polytope. They are interesting in both combinatorics and algebraic geometry.
• Reflexive polytopes correspond to Gorenstein Fano toric varieties. It turns out that there are a finite number of reflexive polytopes in each dimension. These have been classified in low dimension. An introduction by to the reflexive polytopes by Benjamin Nil. Classification of nonsingular 4-dimensional toric Fanos was achieved by Victor Batyrev.
• If X is a Fano variety and H a nonsingular hypersurface in the class -K_X, then H is a Clabi-Yau variety. Batyrev proposed a construction of mirror pairs of Calabi-Yau hypersurfaces in toric Fano varieties corresponding to reflexive polytopes. A short description of this was given by Mattia Talpo.
3. Toric vector bundles.
• Klyachko gave a combinatorial description of toric vector bundles. A self-contained exposition of this, together with historical remarks, can be found in the article by Sam Payne.
• It is not known if every toric variety has a nontrivial toric vector bundle. This problem was studied by Payne. In particular, there is an example of a toric variety that has no nontrivial toric vector bundle of rank at most 3.
4. The homogeneous coordinate ring of toric varieties.
• David Cox generalized the notion of the homogeneous coordinate ring from projective spaces to toric varieties. An interesting combinatorial feature of the construction is as follows. Toric line bundles on a (smooth) projective toric variety correspond to certain polytopes. One can take all these polytopes for different line bundles and stack them together to a form a cone. The semigroup ring of this cone is the homogeneous coordinate ring.
• Homogeneous coordinate rings for general varieties are called Cox rings. These are interesting invariants for combinatorially defined varieties such as blowups of toric varieties, Fano varieties, moduli spaces of curves, etc.
5. Toric varieties as quotients.
• A nonsingular toric variety X can be expressed as the quotient of an open set U in the affine space by a torus action. This is described in more generality in David Cox's paper above. In the nonsingular case, however, one can give a simple description of the quotient map, which is a toric morphism U->X.
• Toric Deligne-Mumford stacks were defined by Borisov, Chen and Smith. These can be viewed as the stacky versions of the qutient construction by Cox. Toric Deligne-Mumford stacks have simple combinatorial descriptions in terms of fans, similar to toric varieties.