On the gonality sequence of smooth curves: normalizations of singular curves in a quadric surface

Abstract

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 $L\in \operatorname{Pic}^{t}(C)$ with h 0(C,L)=r+1. In this paper, for all g≥40805 we construct several examples of smooth curves C of genus g with d 3(C)/3<d 4(C)/4, i.e. for which a slope inequality fails.