triangulations of polytopes and algebraic geometry
The interaction between polyhedral combinatorics and algebraic geometry is a classical theme. Two examples from the 70's, in which each discipline has benefited from the other, are the Bernstein-Kouchnirenko-Khovanski Theorem on the number of roots of a generic sparse system via mixed volumes and the algebraic proofs by Stanley of the Upper Bound Theorem for simplicial spheres and the g -theorem for simplicial polytopes.
toric hilbert scheme tropical geometry