Very ample polarized self maps extend to projective space

Abstract

Let $X$ be a projective variety defined over an infinite field, equipped with a line bundle $L$, giving an embedding of $X$ into $\mb{P}^m$ and let $\varphi :X\to X$ be a morphism such that ${\varphi }^{\ast }L\cong L^{\otimes q},q\ge 2$. Then there exists an integer $r>0$ extending ${\varphi }^r$ to $\mb{P}^m$.