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

TL;DR

This paper establishes a correspondence between Mori dream regions from small modifications of varieties and $ ext{C}^*$-actions on polarized pairs, revealing chamber decompositions and constructing bordisms via $ ext{C}^*$-equivariant birational models.

Contribution

It introduces a novel link between Mori dream regions and $ ext{C}^*$-actions, including chamber decompositions and a method to construct bordisms from given actions.

Findings

Mori dream regions correspond to $ ext{C}^*$-actions on polarized pairs.

Chamber decompositions of Mori dream regions relate to geometric quotients.

Construction of $ ext{C}^*$-bordisms from multiple geometric quotients.

Abstract

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