Surfaces with surjective endomorphisms of any given degree

We present a complete classification of complex projective surfaces X with nontrivial self-maps (i. e. surjective morphisms f:X → X which are not isomorphisms) of any given degree. Our starting point are results contained in Fujimoto (Publ. Res. Inst. Math. Sci. 38(1):33-92, 2005) and Nakayama (Kyushu J. Math. 56(2):433-446, 2002), they provide a list of surfaces that admit at least one nontrivial self-map. By a case by case analysis that blends geometrical and arithmetical arguments, we then exclude that certain prime numbers appear as degrees of nontrivial self-maps of certain surfaces. © 2011 Springer-Verlag.