The canonical ring of a 3-connected curve

TL;DR

This paper proves that the canonical ring of a 3-connected curve with certain properties is generated in degree 1, extending classical results to singular and reducible curves with specific connectivity conditions.

Contribution

It establishes generation in degree 1 of the canonical ring for 3-connected curves with planar singularities, generalizing known results to broader classes of algebraic curves.

Findings

Canonical ring of 3-connected curves is generated in degree 1

Extension of generation results to singular and reducible curves

Conditions on invertible sheaves ensuring generation in degree 1

Abstract

Let C be a projective curve either reduced with planar singularities or contained in a smooth algebraic surface. We show that the canonical ring R(C, \omega_C)= \oplus_{k \geq 0} H^0(C, \omega_C^k is generated in degree 1 if C is 3-connected and not (honestly) hyperelliptic; we show moreover that R(C, L)=\oplus_{k \geq 0} H^0(C,L^k)$ is generated in degree 1 if C is reduced with planar singularities and L is an invertible sheaf such that deg L_{|B} \geq 2p_a(B)+1 for every B \subseteq C.