On The Cohomology of $N_C(-2)$ in Positive Characteristic

TL;DR

This paper extends the understanding of the cohomology of the twisted normal bundle of general Brill--Noether curves in projective 3-space from characteristic zero to positive characteristic, revealing many new exceptions in characteristic 2.

Contribution

It generalizes known results about $H^0(N_C(-2))=0$ from characteristic zero to positive characteristic, especially highlighting new exceptions in characteristic 2.

Findings

In characteristic zero, $H^0(N_C(-2))=0$ with finitely many exceptions.

In positive characteristic, notably characteristic 2, many new exceptions are found.

The results deepen understanding of the geometry of Brill--Noether curves in different characteristics.

Abstract

Let $C \subset \mathbb{P}^3$ be a general Brill--Noether curve. A classical problem is to determine when $H^0(N_C(-2)) = 0$, which controls the quadric section of $C$. So far this problem has only been solved in characteristic zero, in which case $H^0(N_C(-2)) = 0$ with finitely many exceptions. In this note, we extend these results to positive characteristic, uncovering a wealth of new exceptions in characteristic 2.