Generalization of a Max Noether's Theorem

TL;DR

This paper extends Max Noether's Theorem to a broader class of algebraic curves, including nearly Gorenstein and certain non-Gorenstein nonhyperelliptic curves, with independent proofs for each case.

Contribution

It generalizes Max Noether's Theorem to nearly Gorenstein and specific non-Gorenstein curves, providing both extrinsic and intrinsic proofs.

Findings

Max Noether's Theorem holds for nearly Gorenstein curves.

The theorem is valid for integral nonhyperelliptic curves with unibranch non-Gorenstein points.

Two independent proofs are provided: extrinsic and intrinsic.

Abstract

Max Noether's Theorem asserts that if $\ww$ is the dualizing sheaf of a nonsingular nonhyperelliptic projective curve then the natural morphisms $\text{Sym}^nH^0(\omega)\to H^0(\omega^n)$ are surjective for all $n\geq 1$. This is true for Gorenstein nonhyperelliptic curves as well. We prove this remains true for nearly Gorenstein curves and for all integral nonhyperelliptic curves whose non-Gorenstein points are unibranch. The results are independent and have different proofs. The first one is extrinsic, the second intrinsic.