Mori dream bonds and C∗${\mathbb {C}}^*$-actions

We construct a correspondence between Mori dream regions arising from small modifications of normal projective varieties and (Formula presented.) -actions on polarized pairs which are bordisms. Moreover, we show that the Mori dream regions constructed in this way admit a chamber decomposition on which the models are the geometric quotients of the (Formula presented.) -action. In addition, we construct, from a given (Formula presented.) -action on a polarized pair for which there exist at least two admissible geometric quotients, a (Formula presented.) -equivariantly birational (Formula presented.) -variety, whose induced action is a bordism, called the pruning of the variety.