The normal bundle of a general canonical curve of genus at least 7 is semistable

TL;DR

This paper proves that for a general canonical curve of genus at least 7, the normal bundle is semistable, with stability in certain genus congruence cases, advancing understanding of vector bundle stability on algebraic curves.

Contribution

It establishes the semistability of the normal bundle for general canonical curves of genus at least 7, except for specific low-genus cases, and identifies conditions for stability.

Findings

Normal bundle is semistable for genus g ≥ 7, g ≠ 4, 6

Normal bundle is stable when g ≡ 1 or 3 mod 6

Results hold over algebraically closed fields of arbitrary characteristic

Abstract

Let $C$ be a general canonical curve of genus $g$ defined over an algebraically closed field of arbitrary characteristic. We prove that if $g \notin \{4,6\}$, then the normal bundle of $C$ is semistable. In particular, if $g \equiv 1$ or $3$ mod $6$, then the normal bundle is stable.