Curves with canonical models on scrolls

Let [Formula: see text] be an integral and projective curve whose canonical model [Formula: see text] lies on a rational normal scroll [Formula: see text] of dimension [Formula: see text]. We mainly study some properties on [Formula: see text], such as gonality and the kind of singularities, in the case where [Formula: see text] and [Formula: see text] is non-Gorenstein, and in the case where [Formula: see text], the scroll [Formula: see text] is smooth, and [Formula: see text] is a local complete intersection inside [Formula: see text]. We also prove that the canonical model of a rational monomial curve with just one singular point lies on a surface scroll if and only if the gonality of the curve is at most [Formula: see text], and that it lies on a threefold scroll if and only if the gonality is at most [Formula: see text].