The Harder-Narasimhan Filtration of a Trigonal Canonical Curve

TL;DR

This work analyzes the stability of canonical curve normal bundles by exploiting the geometry of scrolls containing special curves. For a general trigonal canonical curve of genus $g$, the main result identifies the destabilizing subbundle $N_{C/S}$ coming from the containing scroll $S$, proving that the Harder-Narasimhan filtration of $N_{C/\mathbb{P}^{g-1}}$ begins with $N_{C/S}$. In the genus $6$ tetragonal case, the paper shows that $0\subset N_{C/S} \subset N_{C/\mathbb{P}^5}$ is the HN-filtration, with $N_{Q/\mathbb{P}^5}|_C$ semistable, and discusses the implications for the broader stability landscape of normal bundles on canonical curves. The approach combines degeneration to unions of rational (or elliptic) curves, adjusted slope stability on nodal curves, and known semistability results for elliptic normal curves, yielding explicit filtrations and paving the way for further exploration of higher gonality cases.

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 $6$.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 2.1
• Proposition 2.2
• Lemma 2.3
• Example 2.4
• Lemma 2.5
• proof
• Definition 2.6
• Proposition 2.7
• Proposition 2.8