Projective embedding of projective spaces (Q1281144)

The author studies embeddings \(\phi:M \rightarrow P\) from a linear space \(M\) in a linear space \(P\). Examples of embeddings with \(\dim M >\dim P\) are given, but the author also focusses on conditions on embeddings such that \(\dim M=\dim P\). He introduces two properties (\textbf{G}) and (\textbf{E}) and he also describes a condition called the \textit{exchange condition}. Linear spaces satisfying conditions (\textbf{G}) and (\textbf{E}) are called \textit{locally complete}. It is then shown that a locally complete embedding of a linear space \(M\) in a linear space \(P\) which satisfies the exchange condition, satisfies \(\dim M=\dim P\). To conclude the article, the author constructs an example of an embedding \(\phi\) of an \(m\)-dimensional Pappian projective space into an \(n\)-dimensional Pappian projective space with \(m>n\) that maps any \(n+1\) independent points onto \(n+1\) independent points, and which, for \(n \geq 3\), satisfies property (\textbf{E}).