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Rational homogeneous spaces as geometric realizations of birational transformations

Gianluca Occhetta, Eleonora A. Romano, Luis E. Solá Conde, Jarosław A. Wiśniewski

TL;DR

The paper develops a framework to realize birational maps as geometric quotients generated by ${\mathbb C}^*$-actions, linking Geometric Invariant Theory with birational geometry on smooth projective varieties. Focusing on equalized ${\mathbb C}^*$-actions on rational homogeneous spaces, it classifies actions with isolated extremal fixed points and constructs explicit Cremona transformations from the induced birational maps on projectivized normal bundles; prominent examples include inversions on quadrics, balanced and isotropic Grassmannians, spinor varieties, and the ${\rm E}_7$ case. By restricting actions to invariant subvarieties and considering rational homogeneous bundles and linear sections, the authors realize additional birational maps of type $(2,1)$ (Fu–Hwang) in homogeneous and non-homogeneous settings, enriching the catalog of geometric realizations and illustrating a representation-theoretic path to birational geometry.

Abstract

A geometric realization of a birational map $ψ$ among two complex projective varieties is a variety $X$ endowed with a $\mathbb{C}^*$-action inducing $ψ$ 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.

Rational homogeneous spaces as geometric realizations of birational transformations

TL;DR

The paper develops a framework to realize birational maps as geometric quotients generated by -actions, linking Geometric Invariant Theory with birational geometry on smooth projective varieties. Focusing on equalized -actions on rational homogeneous spaces, it classifies actions with isolated extremal fixed points and constructs explicit Cremona transformations from the induced birational maps on projectivized normal bundles; prominent examples include inversions on quadrics, balanced and isotropic Grassmannians, spinor varieties, and the case. By restricting actions to invariant subvarieties and considering rational homogeneous bundles and linear sections, the authors realize additional birational maps of type (Fu–Hwang) in homogeneous and non-homogeneous settings, enriching the catalog of geometric realizations and illustrating a representation-theoretic path to birational geometry.

Abstract

A geometric realization of a birational map among two complex projective varieties is a variety endowed with a -action inducing 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 --, by considering -actions on certain rational homogeneous spaces and their subvarieties.
Paper Structure (19 sections, 21 theorems, 77 equations, 2 figures, 7 tables)

This paper contains 19 sections, 21 theorems, 77 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on ${\mathbb C}^*$-actions
  3. Generalities on ${\mathbb C}^*$-actions
  4. Exceptional locus of the induced birational transformation
  5. Local description of ${\mathbb C}^*$-invariant divisors
  6. Torus actions on rational homogeneous spaces
  7. Notation and basic facts on rational homogeneous varieties
  8. ${\mathbb C}^*$-actions on rational homogeneous varieties
  9. The action of a maximal torus on a complete flag manifold
  10. Equalized actions with isolated extremal fixed points on rational homogeneous spaces
  11. Equivariant Cremona transformations
  12. Example: Quadrics
  13. Example: Balanced Grassmannians
  14. Example: Isotropic Grassmannians
  15. Example: Spinor varieties
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\psi:{\mathbb P}(M^\vee_{n\times n}({\mathbb C}))\dashrightarrow {\mathbb P}(M^\vee_{n\times n}({\mathbb C}))$ be the projectivization of the inversion map of $n\times n$ matrices, and let $\psi_s$, $\psi_a$ be its restrictions to the projetivizations of the spaces of symmetric and skew-symmetr

Figures (2)

  • Figure 1: The Dynkin diagram ${\rm E}_6$.
  • Figure 2: A rational homogeneous variety with isolated sink, and the induced action on $\overline{X}(J)$.

Theorems & Definitions (51)

  • Theorem
  • Theorem
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 41 more