A simple remark on holomorphic maps on Torelli space of marked spheres

Abstract

The configuration space of $k\ge 3$ ordered points in the Riemann sphere $\hat{\mathbb{C}}$ is the Torelli space ${\mathcal{U}}_{0,k}$; a complex manifold of dimension $k-3$. If $m,n\ge 4$ and $F:{\mathcal{U}}_{0,m}\to {\mathcal{U}}_{0,n}$ is a non-constant holomorphic map, then we observe that (i) $n\le m$ and (ii) each coordinate of $F$ is given by a cross-ratio.