A Complexity Bound on Faces of the Hull Complex

Given a monomial k[x1, . . . , xn]-module M in the Laurent polynomial ring k[x ±1 1 , . . . , x ±1 n ], the hull complex is defined to be the set of bounded faces of the convex hull of the points {t | x ∈ M} for sufficiently large t. Bayer and Sturmfels [2] conjectured that the faces of this polyhedron are of bounded complexity in the sense that every such face is affinely isomorphic to a subpolytope of the (n − 1)-dimensional permutohedron, which in particular would imply that these faces have at most n! vertices. In this paper, we prove that the latter statement is true, and give a counterexample to the stronger conjecture.