## Projective Normality of Smooth Toric Varieties

As of the last updating of this webpage, it is an open question
whether smooth toric varieties are always projectively normal. This
webpage will record progress and partial results as we look for a
proof or a counterexample.

In August 2007 there was an
Oberwolfach
mini-workshop on this topic. The report for the workshop is available here. It includes a description of the problem.

In April 2009 there was a workshop on this topic and related problems
at the American Institute of Mathematics. See the webpage.

#### Progress since the first workshop

- Haase/Nill/Paffenholz/Santos have shown that the smoothness
hypothesis is not necessary in the result of Fakhruddin (math.AG/0208178).
Explicitly, if P and Q are lattice polygons, where the normal fan of P
refines that of Q, then every lattice point in P+Q is a sum of a lattice point
in P and one in Q. See arxiv:0711.4393.
- Payne's work on Frobenius splitting arxiv/0802.4302, with applications arxiv/0805.1252

#### Progress since the second workshop

- Winfried Bruns has released his code to search for counterexamples for these questions: see the project webpage.

#### Selected bibliography

This is a * very* selected bibliography - please send any other papers you think belong on this list.

- Bruns and Gubeladze
* Semigroup algebras and discrete geometry*,
* Unimodular covers of multiples of polytopes*,
Mathscinetlinks: 1,
2.
- Bruns, Gubeladze and Trung
*Problems and algorithms for affine semigroups,*
MathSci.
- Ein, Lazarsfeld.
* Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension.*
MathSci.
- Fakhruddin
* Multiplication maps of linear systems on smooth projective toric surfaces *. ArXiv.
- Howard
*Matroids and geometric invariant theory of torus actions on flag spaces.*. MathSci.
- Oda
*Problems on Minkowski sums of convex lattice
polytopes*. Unpublished preprint. NEW! Now posted on the Arxiv.
- Ohsugi, Hibi
*Normal polytopes arising from finite graphs.*
MathSci

See also the book by Winfried Bruns and Joseph Gubeladze on * Polytopes, rings and K-theory* available here.

This page is a "static Wiki" - contributions are welcome. Send your
examples/proofs/counterexamples for inclusion in the webpage to me at the email address on my main page

* Site maintained by Diane Maclagan. Last updated December 7, 2007 *

This material is based upon work supported by the National Science
Foundation under Grant No. DMS 0500386.
Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).