The extremal Secant Conjecture for curves of arbitrary gonality

TL;DR

This paper proves the extremal case of the Green-Lazarsfeld Secant conjecture for curves with arbitrary gonality, establishing vanishing of certain Koszul groups under specific generic conditions.

Contribution

It confirms the conjecture in the extremal case for curves of any gonality, expanding previous results to more general curve classes.

Findings

Proves the conjecture for extremal degree cases with generic conditions.

Allows arbitrary gonality, including cases where gonality equals Cliff(C)+2.

Establishes vanishing of Koszul groups in these cases.

Abstract

Let $C$ be a curve and $L$ a very ample line bundle. The Green-Lazarsfeld Secant conjecture predicts that if the degree of $L$ is at least $2g+p+1-2h^1(C,L)-Cliff(C)$ and if, in addition, $L$ is $p+1$ very ample, then the Koszul group $K_{p,2}(C,L)$ vanishes. In this article, we establish the conjecture in the extremal case, i.e.\ the case where the degree is exactly $2g+p+1-2h^1(C,L)-Cliff(C)$, subject to explicit genericity assumptions on $C$ and $L$. In particular, the gonality of $C$ is allowed to be arbitrary (in our cases $gon(C)=Cliff(C)+2$).