2024/06/22 by Konstanze Rietsch, Lauren Williams, Rietsch, Konstanze +1
Social Sciences · #05E14 #14M25 #52B05 #52B20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Political and Social Issues
paper · doi:10.48550/arxiv.2406.15803
openalex publication_date 2024/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points \ei-ej \vert i ≠ j\ ∪ \± ei\ in ℝn, where e1,…,en is the standard basis of ℝn. Such a polytope can be encoded by a quiver Q with vertices V ⊆ \v1,…,vn\ ∪ \⋆\, where each edge vj→ vi or ⋆ → vi or vi→ ⋆ gives rise to the point ei-ej or ei or -ei, respectively; we denote the corresponding polytope as Root(Q). These polytopes have been studied extensively under names such as edge polytope and root polytope. We show that if the quiver Q is strongly-connected then the root polytope Root(Q) is reflexive and terminal; we moreover give a combinatorial description of the facets of Root(Q). We also show that if Q is planar, then Root(Q) is (integrally equivalent to the) polar dual of the flow polytope of the dual quiver. Finally we consider the case that Q comes from a ranked poset P, and show that Root(Q) is polar dual to (a translation of) a marked poset polytope. We then study the toric variety Y(FQ) associated to the face fan FQ of Root(Q). If Q comes from a ranked poset P we give a combinatorial description of the Picard group of Y(FQ), and we show that Y(FQ) is a small partial desingularisation of the Hibi toric variety YO(P) of the order polytope O(P). We show that Y(FQ) has a small crepant toric resolution of singularities Y(\widehatFQ), and as a consequence that the Hibi toric variety YO(P) has a small resolution of singularities for any ranked poset P. These results have applications to mirror symmetry.