Curves and 0-cycles on projective surfaces

LetC be a smooth curve with ag_n¹, i.e. a linear system of dimension 1 and degreen, lying on a smooth projective surfaceS. Let :S P^N be the map associated to the line bundleK_S+[C] and letD be a general divisor of the given linear systemg_n¹. LetV be the linear space spanned by the image ofD through . We study the case in whichn:=dimV=1 and in general we discuss the case in whichn is small. The starting point is an analysis of the adjunction map using Bogomolov-Reider-Serrano techniques; several results from curve theory are also needed.