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

Abstract

We link small modifications of projective varieties with a ${\mathbb{C}}^{\ast }$-action to their GIT quotients. Namely, using flips with centers in closures of Białynicki-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 ${\mathbb{C}}^{\ast }$-action which has no finite isotropy for any point. Examples can be constructed upon homogeneous varieties with a ${\mathbb{C}}^{\ast }$-action associated to short grading of their Lie algebras.

29 pages, 7 figures

References in corpus (1)

• High rank torus actions on contact manifolds

Cited by in corpus (3)

• Rational homogeneous spaces as geometric realizations of birational transformations
• Geometric realizations of birational transformations via ${\mathbb{C}}^{\ast }$-actions
• Toric non-equalized flips associated to ${\mathbb{C}}^{\ast }$-actions