A State Polytope Decomposition Formula

We give a decomposition formula for computing the state polytope of a reducible variety in terms of the state polytopes of its components: if a polarized projective variety X is a chain of subvarieties Xi satisfying some further conditions, then the state polytope of X is the Minkowski sum of the state polytopes of Xi translated by a vector τ, which can be readily computed from the ideal of Xi. The decomposition is in the strongest sense in that the vertices of the state polytope of X are precisely the sum of vertices of the state polytopes of Xi translated by τ. We also give a similar decomposition formula for the Hilbert–Mumford index of the Hilbert points of X. We give a few examples of the state polytope and the Hilbert–Mumford index computation of reducible curves, which are interesting in the context of the log minimal model program for the moduli space of stable curves.