Max Noether Theorem for Singular Curves

TL;DR

This paper extends Max Noether's Theorem to all integral curves, including singular ones, by analyzing the combinatorics involved and building upon previous results for Gorenstein and projectively normal curves.

Contribution

It generalizes Max Noether's Theorem to arbitrary integral curves, covering cases with singularities and non-Gorenstein points, filling a significant gap in the theory.

Findings

Proves surjectivity of natural morphisms for all integral curves.

Extends previous results from Gorenstein to non-Gorenstein cases.

Provides a combinatorial framework for the general case.

Abstract

Max Noether's Theorem asserts that if $\omega$ 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 $n\geq 1$. The result was extended for Gorenstein curves by many different authors in distinct ways. More recently, it was proved for curves with projectively normal canonical models, and curves whose non-Gorenstein points are bibranch at worse. Based on those works, we address the combinatorics of the general case and extend the result for any integral curve.