Surjective rational maps and del Pezzo surfaces

Abstract

We study surjective (not necessarily regular) rational endomorphisms $f$ of smooth del Pezzo surfaces $X$. We prove that under certain natural non\,-\,degeneracy condition $f$ can have degree bigger than $1$ only when $(-K_X^2)>5$. Some structural properties of $f$ in the case $X = \p^2$ are also established.

10 pages, 2 figures; v4: non-degeneracy condition added to the statement of Theorem 1.2