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 $X={\mathbb{P}}^2$ are also established.

11 pages, 2 figures; v5: Definition 2.2 expanded, new example in Remark 2.3 added, corrected misprints