Chow quotients of ${\mathbb{C}}^{\ast }$-actions on convex varieties

In this paper we study the Chow quotient $\mathcal{C}X$ of a convex variety $X$ of Picard number one by the action of a one dimensional torus having no non-trivial finite isotropy. Examples of these actions can be found in the rational homogeneous framework. We prove that the subvariety of $\mathcal{C}X$ parametrizing reducible torus-invariant cycles is a simple normal crossing divisor, we compute the Nef and Mori cones of $\mathcal{C}X$, and its anticanonical divisor.