Degree of the Bicanonical Map of a Surface of General Type

Abstract

Let 5 be a smooth complex projective surface of general type, rt: 5 ?> FPn~l be the map defined by nKs. <&? is called pluricanonical map for n ^ 2. Due to the works by Bombieri and others in the 1970s, the nature of $n is determined except when n = 2 and for some low values of invariants of 5 (see [1] for an up-to-date treatment and references). These maps play an important role in the classification of surfaces of general type. Recently, the remaining case of the bicanonical map $2 for low values of invariants has aroused new research interests, due to the work of Francia [8], and especially the remarkable method of Reider ([15], see also [6], [14], [4]). They have proved the following result (among other things).