Rational homogeneous spaces as geometric realizations of birational transformations

TL;DR

This paper explores how certain birational maps between complex projective varieties can be geometrically realized using $ ext{C}^*$-actions on rational homogeneous spaces, providing new insights into their structure.

Contribution

It introduces a method to realize classic birational maps via $ ext{C}^*$-actions on rational homogeneous spaces, linking birational geometry with group actions.

Findings

Realization of inversion maps through $ ext{C}^*$-actions.

Construction of geometric models for Cremona transformations.

Extension of realizations to special birational transformations of type (2,1).

Abstract

A geometric realization of a birational map $\psi$ among two complex projective varieties is a variety $X$ endowed with a $\mathbb{C}^*$-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 $\mathbb{C}^*$-actions on certain rational homogeneous spaces and their subvarieties.