Curves with Canonical Models on Scrolls

TL;DR

This paper investigates the properties of algebraic curves with canonical models on scrolls, focusing on gonality and singularities, and characterizes when certain monomial curves lie on scrolls based on gonality.

Contribution

It provides new criteria for when rational monomial curves with singularities lie on surface or threefold scrolls, based on gonality thresholds.

Findings

Rational monomial curves with one singular point lie on a surface scroll iff gonality ≤ 3.

Such curves lie on a threefold scroll iff gonality ≤ 4.

Analysis of gonality and singularities for curves on scrolls in specific cases.

Abstract

Let $C$ be an integral and projective curve whose canonical model $C'$ lies on a rational normal scroll $S$ of dimension $n$. We mainly study some properties on $C$, such as gonality and the kind of singularities, in the case where $n=2$ and $C$ is non-Gorenstein, and in the case where $n=3$, the scroll $S$ is smooth, and $C'$ is a local complete intersection inside $S$. We also prove that a rational monomial curve with just one singular point lies on a surface scroll if and only if its gonality is at most $3$, and that it lies on a threefold scroll if and only if its gonality is at most $4$.