A note on embeddings of projective spaces

Abstract

Let $\textbf{k}$ and $\textbf{K}$ be commutative fields, and $l,m$ integers with $l ≥ 1, m ≥ 2$. Suppose that there exists an embedding $\psi$ of $PG(m + l,\textbf{k})$ to $PG(m,\textbf{K})$, then we have $r = dim_{\textbf{k}}\textbf{K} ≥ 4$ and $m ≥ ≤ft [{3l}\over{r-3}\right] - 1$. Conversely, there exists an embedding $\psi$ of $PG(l + m,\textbf{k})$ to $PG(m,\textbf{K})$ if $m ≥ ≤ft [{3l}\over{r-3}\right] - 1$ and if (1) $dim_{\textbf{k}}\textbf{K} = 4$, or (2) $dim_{\textbf{k}}\textbf{K} > 4$ and $\textbf{K}$ is a cyclic extension of $\textbf{k}$ with some additional conditions on l and r