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

TL;DR

This paper investigates conditions under which the recession cones of a polyhedral complex form a fan, providing examples, a positive condition involving Minkowski-Weyl, and applications to toric schemes.

Contribution

It identifies a Minkowski-Weyl type condition ensuring recession cones form a fan and applies this to classify proper toric schemes over valuation rings.

Findings

Recession cones do not always form a fan.

A Minkowski-Weyl condition guarantees the formation of a fan.

Classification of proper toric schemes via polyhedral complexes.

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.