A generalization of Max Noether’s theorem

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 $n\geq 1$. This is true for Gorenstein nonhyperelliptic curves as well. We prove that 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.