On the gonality sequence of smooth curves

Let C be a smooth curve of genus g. For each positive integer r the r-gonality d _r (C) of C is the minimal integer t such that there is \({L\in {\rm Pic}^t(C)}\) with h ⁰(C, L) = r + 1. Here we use nodal plane curves to construct several smooth curves C with d ₂(C)/2 < d ₃(C)/3, i.e., for which a slope inequality fails.