Mori Dream Bonds and ${\mathbb{C}}^{\ast }$-actions

Abstract

We construct a correspondence between Mori dream regions arising from small modifications of normal projective varieties and ${\mathbb{C}}^{\ast }$-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 ${\mathbb{C}}^{\ast }$-action. In addition we construct, from a given ${\mathbb{C}}^{\ast }$-action on a polarized pair for which there exist at least two admissible geometric quotients, a ${\mathbb{C}}^{\ast }$-equivariantly birational ${\mathbb{C}}^{\ast }$-variety, whose induced action is a bordism, called the pruning of the variety.

25 pages. To appear in Math. Nachr