On the canonical ring of curves and surfaces

TL;DR

This paper investigates conditions under which the canonical ring of a curve or surface is generated in low degrees, providing new criteria based on connectivity and hyperellipticity for algebraic curves on surfaces.

Contribution

It establishes new generation results for the canonical ring of curves on surfaces, linking numerical connectivity and hyperellipticity to the degree of generators.

Findings

Canonical ring of a curve is generated in degree 1 if the curve is numerically 4-connected, not hyperelliptic, and even.

On a surface of general type with p_g>0 and q=0, the surface's canonical ring is generated in degree ≤3 under certain curve conditions.

Provides criteria connecting the geometry of curves on surfaces to algebraic properties of their canonical rings.

Abstract

Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic surface. We show that the canonical ring R(C, \omega_C) is generated in degree 1 if C is numerically 4-connected, not hyperelliptic and even (i.e. with K_C of even degree on every component). As a corollary we show that on a smooth algebraic surface of general type with p_g(S)>0 and q(S)=0 the canonical ring R(S, K_S) is generated in degree \leq 3 if there exists a curve C in |K_S| numerically 3-connected and not hyperelliptic.