Surjective rational maps and del Pezzo surfaces

TL;DR

This paper investigates surjective rational endomorphisms of smooth del Pezzo surfaces, establishing conditions under which such maps can have degree greater than one, especially relating to the surface's degree.

Contribution

It provides new criteria linking the degree of surjective rational maps to the del Pezzo surface's degree, and explores structural properties for the case of the projective plane.

Findings

Maps with degree > 1 only occur when (-K_X^2) > 5

Structural properties of endomorphisms on  are characterized

Conditions for non-degenerate surjective rational maps are identified

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.