TITLE: Are there any algebraic geometry theorems that were proved using combinatorics?
QUESTION [19 upvotes]: I'm collaborating with some algebraic geometers in a paper, and when writing the introduction I mentioned the interaction of combinatorics and algebraic geometry, and gave some examples like the combinatorial Nullstellensatz, the affirmative answer to the conjecture of Read and Rota-Heron-Welsh and the graph-theoretic analogue of the Riemann-Roch theorem, but then, it seems that all these interactions are one way.
I would like to know of an important algebraic geometry theorem proved using combinatorics, or at least used combinatorics in a critical part.

REPLY [2 votes]: Disclaimer: I am quite ignorant of algebraic geometry. I am also most opinionated about subjects that I do not know. Caveat lector!
Ehrhart theory is a fundamental tool in toric geometry. In my opinion Ehrhart polynomials and Ehrhart generating functions belong to the land of combinatorics. When Eugene Ehrhart invented these beautiful things, he was doing combinatorics.