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.



Are there any algebraic geometry theorems that were proved using combinatorics?
Jan Draisma's chapter "Noetherianity up to symmetry" in the book *Combinatorial Algebraic Geometry* (Springer LNM 2108) presents various finiteness theorems that are based on Kruskal's tree theorem (or actually the special case known as Higman's lemma).


The rest of the book also contains some potential examples, although there's some risk of getting tangled up in debates about where the combinatorics ends and the algebra or geometry begins.


