Surfaces with nontrivial surjective endomorphisms of any given degree

TL;DR

This paper classifies complex projective surfaces that admit nontrivial self-maps of any degree, combining geometric and arithmetic methods to determine which degrees are possible for such maps.

Contribution

It provides a complete classification of surfaces with nontrivial self-maps of any degree, extending previous partial results with a detailed case analysis.

Findings

Identifies surfaces admitting nontrivial self-maps of specific degrees

Excludes certain prime degrees for self-maps on particular surfaces

Completes the classification of surfaces with arbitrary degree self-maps

Abstract

We present a complete classification of complex projective surfaces $X$ with nontrivial self-maps (i.e. surjective morphisms $f:X\rightarrow X$ which are not isomorphisms) of any given degree. The starting point of our classification are results contained in Fujimoto and Nakayama that provide a list of surfaces that admit at least one nontrivial self-map. We then proceed by a case by case analysis that blends geometrical and arithmetical arguments in order to exclude that certain prime numbers appear as degrees of nontrivial self-maps of certain surfaces.