On Gonality and Canonical Models of Unicuspidal Rational Curves

TL;DR

This paper investigates the gonality and canonical models of rational unicuspidal curves, especially non-Gorenstein ones, classifying them up to genus 6 and deriving formulas for hypersurface spaces.

Contribution

It provides a classification of non-Gorenstein rational unicuspidal curves via gonality and canonical models up to genus 6, extending to general families of curves.

Findings

Classification of non-Gorenstein curves up to genus 6

Derived formula for dimension of hypersurfaces containing the canonical model

Applied formulas to specific cases of rational unicuspidal curves

Abstract

We study the gonality and canonical model of a rational unicuspidal curve C. We are mainly interested in the case where C is non-Gorenstein. We classify such curves via different notions of gonality, and by its canonical model C', up to genus 6. We do it by means of more general families of curves of arbitrary genus. Afterwards, we get a general formula for the dimension of the space of hypersurfaces of a fixed degree containing C', which we apply to some particular cases.