Very ample polarized self maps extend to projective space (Q427761)

This note proves that given a projective variety \(X\) over an infinite field, a morphism \(\phi: X\rightarrow X\), and a very ample line bundle \(L\) such that \(\phi^* L\cong L^{\otimes d}\), there exists an iterate of \(\phi\) that can be extended to a finite morphism on \(\mathbb P^m\), where the embedding \(X\rightarrow \mathbb P^m\) is given by \(L\). As the authors remark, \textit{N. Fakhruddin} [J. Ramanujan Math. Soc. 18, No. 2, 109--122 (2003; Zbl 1053.14025)] has previously proved a related result, where he fixes \(\phi\) and changes \(L\) to a higher tensor power. The article also contains an example where taking an iterate is necessary to obtain the extension.