# Singular rational curves with points of nearly-maximal weight

## TL;DR

This paper investigates rational curves with specific singularities of nearly-maximal weight, classifying them based on linear series, scrolls, and gonality, especially for hyperelliptic types with semigroups of maximal or near-maximal weight.

## Contribution

It provides a partial classification of rational curves with a unique unibranch singularity of hyperelliptic type and maximal or nearly-maximal weight, focusing on their linear series, scrolls, and gonality.

## Key findings

- Classified rational curves with hyperelliptic singularities of maximal or nearly-maximal weight.
- Analyzed the linear series and scrolls supporting these curves.
- Determined their gonality and geometric properties.

## Abstract

In this article we study rational curves with a unique unibranch genus-$g$ singularity, which is of {\it $\ka$-hyperelliptic} type in the sense of \cite{To}; we focus on the cases $\ka=0$ and $\ka=1$, in which the semigroup associated to the singularity is of (sub)maximal weight. We obtain a partial classification of these curves according to the linear series they support, the scrolls on which they lie, and their gonality.