When do the recession cones of a polyhedral complex form a fan?

Abstract

We study the problem of when the collection of the recession cones of a polyhedral complex forms also a complex. We exhibit an example showing that this is no always the case. We also show that if the support of the given polyhedral complex satisfies a Minkowski-Weyl type condition, then the answer is positive. As a consequence, we obtain a classification theorem for proper toric schemes over a discrete valuation ring in terms of complete strongly convex rational polyhedral complexes.