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In light of the nice responses to this question, I wonder what are some open problems in the area of toric geometry? In particular,

What are some open problems relating to the algebraic combinatorics of toric varieties?

and

What are some open problems relating to the algebraic geometry of toric varieties?

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James Davidoff
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  • I wanted to mention the question of existence of full exceptional collections on toric Fano varieties, but this was answered in a very recent paper by Efimov. Probably there are still interesting questions left regarding derived categories of toric varieties... – Piotr Achinger Oct 27 '10 at 22:11
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    Let $X$ be a complete toric variety, necessarily neither smooth nor projective. Is there a nontrivial vector bundle on $X$? Payne has constructed examples that have no one- or two-dimensional bundles, and constructing vector bundles of higher rank on these is still open. The "mirror" question is whether there exists a Lagrangian submanifold of $(\mathbb{C}^*)^n$ satisfying certain asymptotic conditions coming from the fan of $X$. – David Treumann Oct 28 '10 at 01:21
  • Although I don't really follow this, but I think that the following is an open conjecture.

    Conjecture Every ample divisor on a smooth toric variety is very ample and induces a projectively normal embedding.

    Is that right?

     – Karl Schwede Oct 28 '10 at 04:04
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    @Piotr. Having looked at the introduction to the Efimov paper, it seems that there are no new positive results there about full exceptional collections. Kawamata showed there is always a full exceptional collection of coherent sheaves on a smooth toric DM stack. And Orlov (says Efimov) conjectures that there should exist a strong full exceptional collection. It was conjectured that toric Fano varieties should admit full exceptional collections of line bundles. Efimov shows that this is false in general.

    Apologies to Efimov if I read him incorrectly.

     – Chris Brav Oct 28 '10 at 07:59
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    @Karl: ample $\Rightarrow$ very ample is known for smooth toric varieties, but ample $\Rightarrow$ projectively normal is indeed a well-known open question. (I say question'' rather thanconjecture'' as it doesn't seem like all experts believe it.) – Arend Bayer Oct 28 '10 at 12:34
  • @Chris: I know, my "this was answered" wasn't intended to mean "this was answered YES".

    @AByer: both statements would follow from the existence of a diagonal Frobenius splitting of the toric variety in question (at least in positive characteristic, and maybe in characteristic 0 using the "toric" Frobenius morphism). Are there counterexamples to this?

     – Piotr Achinger Oct 28 '10 at 14:50
  • @Arend: Thanks, that's right, and also thanks for pointing out that it's just an open question and not a conjecture (I didn't know what people's feeling on this). @Piotr: I thought the existence of diagonal Frobenius splittings on toric varieties was shown to have a flaw, see Sam Payne's paper on the same subject. The point I thought was that the diagonal is not a torus invariant subvariety and so the canonical Frobenius splitting doesn't compatibly split it (one has to choose a more clever splitting, and Sam analyzes when they exist). – Karl Schwede Oct 28 '10 at 17:43

2 Answers2

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My favourite is Oda's Strong Factorization Conjecture:

Can a proper, birational map between smooth toric varieties be factored as a composition of a sequence of smooth toric blow-ups followed by a sequence smooth toric blow-downs?

Note that if you are allowed to intermingle the blow-ups and blow-downs (the weak version) it has been proved. In fact, it was proved for general varieties in characteristic 0 using the toric case:

Torification and Factorization of Birational Maps. Abramovich, Karu, Matsuki, Wlodarczyk.

A conjectural algorithm for computing toric strong factorizations can be found in the following arXiv article:

On Oda's Strong Factorization Conjecture. Da Silva, Karu.

Alexander Duncan
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The following question was posed by Rikard Bögvad in the paper On the homogeneous ideal of a projective nonsingular toric variety:

Is the toric ideal of a smooth projectively normal toric variety generated by quadrics?

This is interesting, since toric ideals have an explicit description. In particular, it is not known if the coordinate ring of a smooth projectively normal toric variety is Koszul. Smoothness is of course essential here, since there are many toric hypersurfaces of degree $\ge 3$, e.g., $x_0^n=x_1 \cdots x_n$.

J.C. Ottem
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    I saw that this paper by Bögvad was withdrawn due to an error. In view of this, does the stated question still make sense/is interesting? – Yellow Pig Jan 12 '20 at 23:05