Sheaf Cohomology of the Moduli Space of Spatial Polygons and Lattice Points

Abstract

Let $\mathcal{M}_n$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Let $K_{n}^{*}=\Lambda^{n-3}T\mathcal{M}_n$ be the dual bundle of the canonical bundle on $\mathcal{M}_n$. In this paper we determine the sheaf cohomology $H^*(\mathcal{M}_n,K_{n}^{*})$. We have $H^q(\mathcal{M}_n,K_{n}^{*})=0$ $(q\geq 1)$ and $\dim H^0(\mathcal{M}_n,K_{n}^{*})$ is equal to the number of lattice points in the convex polytope $\Delta_n$ in $\mathbf{R}^{n-3}$.