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I wonder whether there are problems whose statement do not mention toric varieties (nor simple polytopes, vanishing sets of binomials, etc.), but whose proof nicely and essentially uses them?

To give an idea, two simple random examples -- not for toric varieties, but for other concepts:

  • a one-line computation of $H^*(K/T)$, the cohomology of flag variety, using equivariant cohomology, in an answer MSO:21670 by Allen Knutson;
  • the hard Lefschetz theorem, that for a Kähler $X$, the map is $\omega^{n-i}: H^i(X) \to H^{2n-i}(X)$ is an isomorphism; the now standard proof by Chern uses representation theory of $sl_2$.
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  • $\begingroup$ (Sorry, I had an answer about polytopes but now I see you want to exclude them.) $\endgroup$ – Sam Hopkins Mar 4 '17 at 23:28
  • $\begingroup$ Not really, but sort of:mathoverflow.net/questions/52585/… $\endgroup$ – Bertie Mar 4 '17 at 23:33
  • $\begingroup$ The GIT quotient of the conjugation action of an adjoint group on its wonderful compactification equals the quotient of the Weyl group action ofnthe toric compactification of a maximal torus. That is a little bit surprising. $\endgroup$ – Jason Starr Mar 5 '17 at 0:06
  • $\begingroup$ @JasonStarr, could you explain what you mean or give a reference, please? $\endgroup$ – evgeny Mar 5 '17 at 6:50
  • $\begingroup$ @evgeny "could you explain what you mean" Yes. Every group of adjoint type acts on itself by conjugation, and that action extends to wonderful compactification of that adjoint group. It is a little bit surprising that the analysis of the "Geometric Invariant Theory" quotient of the wonderful compactification by this positive-dimensional group reduces to the action of a finite group on a particular toric variety coming from a maximal torus and the root system. $\endgroup$ – Jason Starr Mar 5 '17 at 7:57
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There are lots of applications of toric varieties to singularities, e.g., the proof of the weak factorization theorem in characteristic zero. (Indeed, the name of the linked paper is "Torification and factorization of birational maps.") The weak factorization theorem states roughly that a birational map between (possibly singular) varieties can be factored into blow-ups and blow-downs.

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  • $\begingroup$ Thank you, dhy! There are also nice slides on weak factorization with many images by J. Wlodarczyk and lectures by K. Matsuki, whose paragraph 0 briefly explaines the plan. $\endgroup$ – evgeny Mar 27 '17 at 7:47
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To any matroid $M$ on ground set $E$ we associate the characteristic polynomial $$p_M(t) = \sum_{A \subseteq E} (-1)^{|A|}t^{r(E) - r(A)}$$ where $r(A)$ denotes the rank of $A$. Let $r(E) = r.$ It is known that $$p_M(t) = a_0 t^r - a_1 t^{r-1} + \cdots + (-1)^r a_r$$ for $a_i \geq 0$ (i.e. coefficients alternate in sign). Moreover the coefficients are known to have some combinatorial interpretation. It was a long standing conjecture of Rota-Heron-Welsh that the sequence $(a_0, a_1, \cdots, a_r)$ is log-concave.

In Log-concavity of characteristic polynomials and the Bergman fan of matroids, Huh and Katz show that the Rota-Heron-Welsh conjecture is true for "realizable" matroids by making use of intersection theory and toric varieties. Adiprasito, Huh, and Katz prove the conjecture for general matroids in Hodge Theory for Combinatorial Geometries.

Edit: I have found a paper Matroid theory for algebraic geometers by Katz which gives a expository view of the work of Huh and Katz mentioned above as well as more background. It looks like a nice source which may be of interest to viewers of this question; so, I will share it here.

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Since you did not mention you wanted applications to mathematical problems, I allow myself to take examples from other sciences, for instance biology.

The following problem : "On a given position in DNA string, what is the probability that a human has $C$, a gorilla has $C$ and a guenon has $A$?"

These probabilities depends from a large space of paramaters which corresponds to all the possible mutations in the pylogenetic tree that relates the guenon, the gorilla and the human. Hence we get a map from the spaces of mutations in the pylogenetic tree to the space of probability distributions of the 4 characters ($A,T,G,C$) in the DNA sequence under consideration. An open problem in mathematical biology is to describe the image of this map.

Apparently, some hypothesis made by biologists and statisticians allow to show that the (closure) of the image of the above map is a toric variety. And so toric geometry allows to make some progress in the above problem .

I refer to the recent preprint of Michalek (section 11) : https://arxiv.org/pdf/1702.03125.pdf and references in there for more details on these questions.

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There seems to be a long tradition of studying the semi-stable reduction problem via reduction to toric geometry, dating at least to the monograph of Kempf, Knudsen, Mumford, and Saint-Donat.

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  • $\begingroup$ That monograph invented toric geometry, so I guess that bound will be hard to improve. :) $\endgroup$ – Bertie Mar 5 '17 at 22:24

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