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
The topics below are very general. Each topic has enough material
for several talks.
- 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:
Reflexive polytopes. Reflexive polytopes are lattice polytopes
whose polar dual is also a lattice polytope. They are
interesting in both combinatorics and algebraic geometry.
- 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
- 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
Toric vector bundles.
- 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
- 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
The homogeneous coordinate ring of toric varieties.
- 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.
- 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
Toric varieties as quotients.
- 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.
- 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
- 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.