Let C be an integral and projective curve; and let C be its canonical model. We study the relation between the gonality of C and the dimension of a rational normal scroll S where C can lie on. We are mainly interested in the case where C is singular, or even non-Gorenstein, in which case C C. We first analyze some properties of an inclusion C ⊂ S when it is induced by a pencil on C. Afterwards, in an opposite direction, we assume C lies on a certain scroll, and check some properties C may satisfy, such as gonality and the kind of its singularities. At the end, we prove that a rational monomial curve C has gonality d if and only if C lies on a (d − 1)-fold scroll.

Let C be an integral and projective curve; and let C ′ be its canonical model. We study the relation between the gonality of C and the dimension of a rational normal scroll S where C ′ can lie on. We are mainly interested in the case where C is singular, or even non-Gorenstein, in which caseWe first analyze some properties of an inclusion C ′ ⊂ S when it is induced by a pencil on C. Afterwards, in an opposite direction, we assume C ′ lies on a certain scroll, and check some properties C may satisfy, such as gonality and the kind of its singularities. At the end, we prove that a rational monomial curve C has gonality d if and only if C ′ lies on a (d − 1)-fold scroll.