Resolutions of General Canonical Curves on Rational Normal Scrolls

TL;DR

This paper investigates the structure of the relative canonical resolution of a general canonical curve on a rational normal scroll, revealing conditions under which the bundle of quadrics becomes unbalanced based on Brill-Noether theory.

Contribution

It provides a criterion for when the bundle of quadrics in the relative canonical resolution is unbalanced, linking geometric properties to Brill-Noether numbers.

Findings

Bundle of quadrics is unbalanced if and only if a0>0 and a specific quadratic inequality holds.

The study connects the unbalanced condition to the Brill-Noether number a0.

Results apply to general curves with certain genus and gonality conditions.

Abstract

Let $C\subset \mathbb{P}^{g-1}$ be a general curve of genus $g$ and let $k$ be a positive integer such that the Brill-Noether number $\rho(g,k,1)\geq 0$ and $g > k+1$. The aim of this short note is to study the relative canonical resolution of $C$ on a rational normal scroll swept out by a $g^1_k=|L|$ with $L\in W^1_k(C)$ general. We show that the bundle of quadrics appearing in the relative canonical resolution is unbalanced if and only if $\rho>0$ and $(k-\rho-\frac{7}{2})^2-2k+\frac{23}{4}>0$.