Geodesic
Root, flow and order polytopes with connections to toric geometry
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e194

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In this paper, we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb {R}^n$, where $e_1,\dots ,e_n$ is the standard basis of $\mathbb {R}^n$. Such a polytope can be encoded by a quiver Q with vertices $V \subseteq \{{\upsilon }_1,\dots ,{\upsilon }_n\} \cup \{\star \}$, where each edge ${\upsilon }_j\to {\upsilon }_i$ or $\star \to {\upsilon }_i$ or ${\upsilon }_i\to \star $ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$, respectively; we denote the corresponding polytope as $\operatorname {Root}(Q)$. 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 $\operatorname {Root}(Q)$ is reflexive and terminal; we moreover give a combinatorial description of the facets of $\operatorname {Root}(Q)$. We also show that if Q is planar, then $\operatorname {Root}(Q)$ is (integrally equivalent to) the polar dual of the flow polytope of the planar dual quiver $Q^{\vee }$. Finally, we consider the case that Q comes from the Hasse diagram of a finite ranked poset P and show in this case that $\operatorname {Root}(Q)$ is polar dual to (a translation of) a marked order polytope. We then go on to study the toric variety $Y(\mathcal {F}_Q)$ associated to the face fan $\mathcal {F}_Q$ of $\operatorname {Root}(Q)$. If Q comes from a ranked poset P, we give a combinatorial description of the Picard group of $Y(\mathcal {F}_Q)$, in terms of a new canonical ranked extension of P, and we show that $Y(\mathcal {F}_Q)$ is a small partial desingularisation of the Hibi projective toric variety $Y_{\mathcal {O}(P)}$ of the order polytope $\mathcal {O}(P)$. We show that $Y(\mathcal {F}_Q)$ has a small crepant toric resolution of singularities $Y(\widehat {\mathcal {F}}_Q)$ and, as a consequence that the Hibi toric variety $Y_{\mathcal {O}(P)}$ has a small resolution of singularities for any ranked poset P. These results have applications to mirror symmetry [61]. 
Rietsch, Konstanze; Williams, Lauren Kiyomi. Root, flow and order polytopes with connections to toric geometry. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e194. doi: 10.1017/fms.2025.10056
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