On the Normal Sheaf of Gorenstein Curves

TL;DR

This paper proves that tetragonal Gorenstein integral curves are complete intersections in rational normal scrolls and analyzes the stability and degree of their normal sheaves, including singular cases and deformation behavior.

Contribution

It establishes the complete intersection property of tetragonal Gorenstein curves in rational normal scrolls and computes the normal sheaf degree for singular curves, extending classical local theory results.

Findings

Normal sheaf is unstable for smooth scrolls when genus ≥ 5.

Degree of the normal sheaf relates to Tjurina and Deligne numbers.

Semicontinuity of the normal sheaf degree under deformations.

Abstract

We show that any tetragonal Gorenstein integral curve is a complete intersection in its respective $3$-fold rational normal scroll S, implying that the normal sheaf on $C$ embedded in S, and in $\mathbb{P}^{g-1}$ as well, is unstable for $g\geq 5$, provided that $S$ is smooth. We also compute the degree of the normal sheaf of any singular reduced curve in terms of the Tjurina and Deligne numbers, providing a semicontinuity of the degree of the normal sheaf over suitable deformations, revisiting classical results of the local theory of analytic germs.