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](https://mathoverflow.net/a/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 $sl\_2$.



Application of toric varieties for problems that do not mention them
There are lots of applications of toric varieties to singularities, e.g., the [proof](https://www.math.ubc.ca/~karu/papers/semist.pdf) 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.


