Rational homogeneous spaces as geometric realizations of birational transformations

Abstract

A geometric realization of a birational map $\psi$ among two complex projective varieties is a variety $X$ endowed with a ${\mathbb{C}}^{\ast }$-action inducing $\psi$ as the natural birational map among two extremal geometric quotients. In this paper we study geometric realizations of some classic birational maps --inversion maps, special Cremona transformations, special birational transformations of type $(2,1)$--, by considering ${\mathbb{C}}^{\ast }$-actions on certain rational homogeneous spaces and their subvarieties.

28 pages, 2 figures

References in corpus (2)

• Small modifications of Mori dream spaces arising from ${\mathbb{C}}^{\ast }$-actions
• Small bandwidth ${\mathbb{C}}^{\ast }$-actions and birational geometry