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

TL;DR

This paper explores the relationship between gonality and the embedding of non-Gorenstein curves' canonical models into rational normal scrolls, establishing a characterization for rational monomial curves.

Contribution

It provides new insights into how the gonality of singular, non-Gorenstein curves relates to their canonical models' placement on rational normal scrolls.

Findings

Gonality of rational monomial curves equals the dimension of the scroll minus one.

Properties of canonical models are linked to the gonality and singularities of the original curve.

Characterization of when a canonical model lies on a rational normal scroll.

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.