Max Noether's Theorem for integral curves

Abstract

We study a celebrated result of Max Noether on global sections of the n-dualizing sheaf of a smooth nonhyperelliptic curve in the case where the curve is integral. We reduce the proof of the statement in such a case to a purely numerical condition, which we show that holds if the non-Gorenstein points are bibranch at worst. This is our main result. We also extend the notion of a canonical embedding for integral curves with unibranch non-Gorenstein points at worst, in a way that we can express the dimensions of the components of the ideal in terms of the main invariants of the curve as well. Afterwards we focus on gonality, Clifford index and Koszul cohomology of non-Gorenstein curves by allowing torsion free sheaves of rank 1 in their definitions. We find an upper bound for the gonality, which agrees with Brill–Noether's one for a rational and unibranch curve. We characterize curves of genus 5 with Clifford index 1, and, finally, we study Green's conjecture for a certain class of curves, called nearly Gorenstein.