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.