On gonality, scrolls, and canonical models of non-Gorenstein curves

Abstract

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 $$C'\not \cong C$$ . We first analyze some properties of an inclusion $$C'\subset S$$ 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.