Georgia Institute of Technology College of Sciences

State polytopes in commutative algebra can be used to detect the geometric invariant theory (GIT) stability of points in the Hilbert scheme. I will review the construction of state polytopes and their role in GIT, and present recent work with Ian Morrison in which we use state polytopes to estabilish stability for curves of small genus and low degree, confirming predictions of the minimal model program for the moduli space of curves.