The Harder-Narasimhan Filtration of a Trigonal Canonical Curve

TL;DR

This paper computes the Harder-Narasimhan filtration of the normal bundle for general trigonal canonical curves and genus 6 canonical curves, revealing their stability properties using geometric embeddings.

Contribution

It provides explicit calculations of the Harder-Narasimhan filtration for these classes of canonical curves, connecting geometric properties with bundle stability.

Findings

Filtration computed for trigonal canonical curves on rational normal surface scrolls.

Explicit filtration determined for genus 6 canonical curves.

Results link geometric embeddings to stability of normal bundles.

Abstract

A trigonal canonical curve lies on a rational normal surface scroll $Q \subset \mathbb{P}^{g-1}$. In this note we use this fact to compute the Harder-Narasimhan filtration of the normal bundle of a general such curve $C$ in $\mathbb{P}^{g-1}$. We also compute the Harder-Narasimhan filtration of the Normal bundle of a general canonical curve of genus $6$.