Algebraic curves violating the slope inequalities

The gonality sequence ( dr )r 1 of a curve of genusg encodes, for < g, important information about the divisor theory of the curve. Mostly i 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” m ay occur in it. We first determine all integersg > 0 such that there is no such jump, for all curves of genusg. Secondly, we compute the leading numbers (up to r D 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.