Chamber Decompositions of Moment Polytopes for Torus Actions of Positive Complexity

Abstract

The present work develops the results of the series of papers by Buchstaber and Terzić on the standard actions of the compact torus $T^n = (S^1)^n$ on the complex Grassmann manifolds $G_{n,2}$. In those works, a hyperplane arrangement in $\mathbb{R}^n$ was introduced that determines the chamber decomposition of the hypersimplex $Δ_{n,2}$ for the $T^n$-action on $G_{n,2}$. We introduce a notion of admissible graph for the standard action of the torus $T^n$ on the complex Grassmannian $G_{n,2}$. In terms of admissible graphs, we give a complete inductive description (with respect to $n \ge 4$) of the admissible polytopes in $Δ_{n,2}$, as well as of the toric varieties arising as closures of $(\mathbb{C}^*)^n$-orbits on $G_{n,2}$ under the standard $(\mathbb{C}^*)^n$-action. We consider the $T^n$-equivariant Plücker embedding $G_{n,2} \hookrightarrow \mathbb{C}P^{N_2}$, where $N_2 = \binom{n}{2}-1$. Using admissible graphs, for the considered $T^n$-actions, we describe hyperplane arrangements in $\mathbb{R}^n$ that determine the chambers in $Δ_{n,2}$ for the $T^n$-actions on $G_{n,2}$ and $\mathbb{C}P^{N_2}$. Gel'fand, Kapranov, and Zelevinsky introduced the notions of secondary polytopes and secondary fans in connection with the problem of describing triangulations of a given convex polytope, which is closely related to the Newton polytopes of discriminants and resultants. For the $T^n$-action on $\mathbb{C}P^{N_2}$, we show that the cones in $\mathbb{R}^n$ with vertex at the origin spanned by the chambers form the secondary fan of the cone spanned by the vertices of $Δ_{n,2}$.