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

* Edoardo Ballico *

**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 *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.