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On embedded products of Grassmannians

Hans Havlicek, Corrado Zanella

Abstract

Let $Γ'$ and $Γ$ be two Grassmannians. The standard embedding $φ:Γ'\timesΓ\to \bar{P}$ is obtained by combining the Plücker and Segre embeddings. Given a further embedding $η: Γ'\timesΓ\to P'$, we find a sufficient condition for the existence of $α\in Aut(Γ)$ and of a collineation $ψ: \bar{P} \to P'$ such that $η=({\rm id}_{Γ'}\timesα)φψ$.

On embedded products of Grassmannians

Abstract

Let and be two Grassmannians. The standard embedding is obtained by combining the Plücker and Segre embeddings. Given a further embedding , we find a sufficient condition for the existence of and of a collineation such that .

Paper Structure

This paper contains 4 sections, 3 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Bilinear g-maps

Key Result

PROPOSITION 2.1

Let $\Sigma'$ and $\Sigma"$ be two semilinear spaces, and $F\,:\,\Sigma'\times\Sigma" \rightarrow\hbox{$\mathbb P$}'$ a right embedding. Assume that $\Sigma"$ is universally embedded in a projective space $\hbox{$\mathbb P$}$ of dimension $n$. Let $\ell$ be a line of $\Sigma'$ and $\ell_1$, $\ell_2$ then for $i=1,2$, $(\ell\times\ell_i)F$ is a hyperbolic quadric of the threedimensional subspace $U

Theorems & Definitions (3)

  • PROPOSITION 2.1
  • PROPOSITION 2.2
  • THEOREM 2.3