Max Noether Theorem for Singular Curves

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.