Small modifications of Mori dream spaces arising from ${\mathbb C}^*$-actions

TL;DR

This paper explores how small modifications of Mori dream spaces with ${ m C}^*$-actions relate to GIT quotients, using flips centered on Białynicki-Birula cells to describe birational transformations.

Contribution

It introduces a systematic approach to construct and describe equivariant birational modifications of Mori dream spaces via flips associated with ${ m C}^*$-actions, linking them to GIT quotients.

Findings

Describes modifications for blowups at sink and source points.

Provides a complete structure for certain smooth varieties with ${ m C}^*$-actions.

Constructs examples from homogeneous varieties with short grading.

Abstract

We link small modifications of projective varieties with a ${\mathbb C}^*$-action to their GIT quotients. Namely, using flips with centers in closures of Bia{\l}ynicki-Birula cells, we produce a system of birational equivariant modifications of the original variety, which includes those on which a quotient map extends from a set of semistable points to a regular morphism. The structure of the modifications is completely described for the blowup along the sink and the source of smooth varieties with Picard number one with a ${\mathbb C}^*$-action which has no finite isotropy for any point. Examples can be constructed upon homogeneous varieties with a ${\mathbb C}^*$-action associated to short grading of their Lie algebras.