On Gonality, Scrolls, and Canonical Models of Non-Gorenstein Curves

Abstract

Let $C$ be an integral and projective curve; and let $C^{\prime }$ be its canonical model. We study the relation between the gonality of $C$ and the dimension of a rational normal scroll $S$ where $C^{\prime }$ can lie on. We are mainly interested in the case where $C$ is singular, or even non-Gorenstein, in which case $C^{\prime }\ncong C$. We first analyze some properties of an inclusion $C^{\prime }\subset S$ when it is induced by a pencil on $C$. Afterwards, in an opposite direction, we assume $C^{\prime }$ 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 $C^{\prime }$ lies on a $(d-1)$-fold scroll.