On linear morphisms of product spaces

TL;DR

This paper investigates conditions under which a linear morphism of a product of projective spaces can be decomposed into an automorphism and a linear morphism after Segre embedding, enhancing understanding of morphism structures.

Contribution

It provides sufficient conditions for expressing a linear morphism of product spaces as a composition involving automorphisms and linear morphisms post-Segre embedding.

Findings

Established conditions for morphism decomposition

Connected morphisms with automorphisms and linear maps

Enhanced understanding of product space morphisms

Abstract

Let $\chi$ be a linear morphism of the product of two projective spaces $PG(n,F)$ and $PG(m,F)$ into a projective space. Let $\gamma$ be the Segre embedding of such a product. In this paper we give some sufficient conditions for the existence of an automorphism $\alpha$ of the product space and a linear morphisms of projective spaces $\phi$, such that $\gamma\phi=\alpha\chi$.