Tetragonal Canonical Curves

TL;DR

The paper determines the Harder-Narasimhan filtration of the normal bundle $N_{C/\mathbb{P}^{g-1}}$ for a general tetragonal canonical curve $C$, showing that the destabilizing subbundle is $N_{C/Q}$ when $g$ is odd and $N_{C/Y}$ when $g$ is even. It leverages the scroll geometry $Q$ containing $C$, the complete-intersection description $C=Y\cap Z$ with divisors $Y$ and $Z$, Schreyer's resolution, and the balance of the first syzygy bundle to identify destabilizing subbundles. The stability of the quotient $N_{Q/\mathbb{P}^{g-1}}|_C$ is established via a degeneration of $C$ to a union of a rational normal curve and an elliptic normal curve, yielding precise bounds on subbundle slopes and thereby the HN-filtration (Theorem T1 for odd $g$ and Theorem T2 for even $g$). These results extend the understanding of stability phenomena for canonical curves beyond the general case, providing explicit filtrations tied to tetragonal geometry and syzygy data.

Abstract

We compute the Harder-Narasimhan Filtration of the normal bundle $N_{C/\mathbb{P}^{g-1}}$ where $C$ is a general tetragonal canonical curve of genus $g$.

Theorems & Definitions (21)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• proof
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4: CS25
• Lemma 2.5