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On Picard Number and Dimension of Algebraic Homogeneous Spaces

Ivan Beldiev, Dmitry Timashev

Abstract

An algebraic variety $X$ is called a homogeneous space if there exists a transitive regular action of an algebraic group on $X$. We prove inequalities between the dimension of a homogeneous space of a linear algebraic group and its Picard number.

On Picard Number and Dimension of Algebraic Homogeneous Spaces

Abstract

An algebraic variety is called a homogeneous space if there exists a transitive regular action of an algebraic group on . We prove inequalities between the dimension of a homogeneous space of a linear algebraic group and its Picard number.

Paper Structure

This paper contains 3 sections, 9 theorems, 18 equations.

Table of Contents

  1. The Picard number of a homogeneous space
  2. Affine homogeneous spaces of reductive groups
  3. Homogeneous spaces of linear algebraic groups: the general case

Key Result

Proposition 1.1

Let $X$ be an irreducible homogeneous space of a linear algebraic group. Then the inequality $\rho(X) \leq \dim X$ holds.

Theorems & Definitions (22)

  • Proposition 1.1
  • Lemma 1.2
  • proof
  • proof : Proof of Proposition \ref{['gen_ieq_old']}
  • Example 1.3
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • ...and 12 more