Curves and \(0\)-cycles on projective surfaces (Q1336208)

Let \(S\) be a smooth projective surface containing a smooth curve \(C\) endowed with a linear series of dimension 1 and degree \(n\). The points of any divisor in that series are easily seen to be mapped by the rational map \(f\) defined by \(|K_S + C |\) to linearly dependent points, thus spanning a linear space of dimension \(n' < n - 1\). The case \(n'\) small (especially \(n' = 1)\) is studied via the adjunction map \(f\) using the Bogomolov-Reider techniques. On top of other results, the authors show that the property \(n' = 1\) essentially leads to the surfaces whose hyperplane sections are either trigonal or hyperlliptic curves, if \(C\) is very ample.