Algebraic curves violating the slope inequalities

Abstract

The gonality sequence $(d_{r})_{r \ge 1}$ of a curve of genus $g$ encodes, for $r<g$, important information about the divisor theory of the curve. Mostly it is very difficult to compute this sequence. In general it grows rather modestly (made precise below) but for curves with special moduli some ``unexpected jumps'' may occur in it. We first determine all integers $g>0$ such that there is no such jump, for all curves of genus $g$. Secondly, we compute the leading numbers (up to $r=19$) in the gonality sequence of an extremal space curve, i.e. of a space curve of maximal geometric genus w.r.t. its degree.