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On linear morphisms of product spaces

Alessandro Bichara, Hans Havlicek, Corrado Zanella

TL;DR

Some sufficient conditions for the existence of an automorphism of the product space and a linear morphism of projective spaces @f, such that @c@f=@a@g, are given.

Abstract

Let $χ$ be a linear morphism of the product of two projective spaces $PG(n,F)$ and $PG(m,F)$ into a projective space. Let $γ$ be the Segre embedding of such a product. In this paper we give some sufficient conditions for the existence of an automorphism $α$ of the product space and a linear morphisms of projective spaces $φ$, such that $γφ=αχ$.

On linear morphisms of product spaces

TL;DR

Some sufficient conditions for the existence of an automorphism of the product space and a linear morphism of projective spaces @f, such that @c@f=@a@g, are given.

Abstract

Let be a linear morphism of the product of two projective spaces and into a projective space. Let be the Segre embedding of such a product. In this paper we give some sufficient conditions for the existence of an automorphism of the product space and a linear morphisms of projective spaces , such that .

Paper Structure

This paper contains 3 sections, 11 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. The first decomposition of $\chi$
  3. The case $\hbox{\rm rad}_{2}\chi\neq\emptyset$

Key Result

Proposition 1.1

Let $\hbox{\rm rad}_{i}\chi$ and $D_i$ be complementary subspaces of $\hbox{\sf I P}_i$ ($i=1,2$). Let $\pi_i\,:\,\hbox{\sf I P}_i \rightarrow D_i$ be the projection onto $D_i$ from $\hbox{\rm rad}_{i}\chi$, and $\chi':=\chi\hbox{$|D_1\times D_2$}$. Then, for every $(X,Y)\in{\cal P}_1\times{\cal P}_

Theorems & Definitions (11)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Theorem 2.8
  • ...and 1 more