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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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  • $\begingroup$ 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... $\endgroup$
    – Piotr Achinger
    Commented Oct 27, 2010 at 22:11
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    $\begingroup$ 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 $X$. $\endgroup$
    – David Treumann
    Commented Oct 28, 2010 at 1:21
  • $\begingroup$ 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? $\endgroup$
    – Karl Schwede
    Commented Oct 28, 2010 at 4:04
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    $\begingroup$ @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. $\endgroup$
    – Chris Brav
    Commented Oct 28, 2010 at 7:59
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    $\begingroup$ @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 than conjecture'' as it doesn't seem like all experts believe it.) $\endgroup$
    – Arend Bayer
    Commented Oct 28, 2010 at 12:34

2 Answers 2

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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.

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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., $x_0^n=x_1 \cdots x_n$.

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    $\begingroup$ 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? $\endgroup$
    – Yellow Pig
    Commented Jan 12, 2020 at 23:05

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