Stability of normal bundles of Brill-Noether curves

TL;DR

This paper investigates the stability properties of the normal bundles of general Brill-Noether curves in projective space, establishing conditions under which these bundles are semistable or stable, with implications for the finiteness of exceptions.

Contribution

It provides new criteria for the semistability and stability of normal bundles of Brill-Noether curves, including explicit bounds on genus and degree, extending understanding of their geometric properties.

Findings

Normal bundle is semistable for certain genus and degree conditions.

Normal bundle is stable under slightly stronger bounds.

Finitely many exceptions exist for non-semistability and non-stability.

Abstract

We prove that the normal bundle of a general Brill-Noether curve of genus $g \geq 1$ and degree $d$ in $\mathbb{P}^r$ is semistable if $g=1$ or $g\geq \left \lceil \frac{5r}{2}\right\rceil r(r-1)$, or $d$ is larger than an explicit function of $g$ and $r$. We further prove that the normal bundle is in fact stable if $g\geq 2$ and either $g$ or $d$ satisfy slightly stronger bounds. In particular, for each $r$ and $g\geq 1$ (respectively, $g\geq2$), there are at most finitely many $(d,g)$ for which the normal bundle of the general Brill-Noether curve is not semistable (respectively, stable).