THE TORIC GEOMETRY OF TRIANGULATED POLYGONS IN EUCLIDEAN SPACE
Benjamin Howard, Christopher Manon, John J. Millson · 2008
Speyer and Sturmfels [SpSt] associated Gröbner toric degenerations Gr2(Cn) T of Gr2(Cn) to each trivalent tree T with n leaves. These degenerations induce toric degenerations MT r of Mr, the space of n ordered, weighted (by r) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers as stratified symplectic spaces and describe the action of the compact part of the torus as “bendings of polygons. ” We prove the conjecture of Foth and Hu [FH] that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida [KY].