On Clifford's theorem for singular curves

TL;DR

This paper extends Clifford's theorem to certain singular algebraic curves, establishing bounds on sections of subcanonical clusters and characterizing cases of equality, thus broadening classical results to more complex curve types.

Contribution

It generalizes Clifford's theorem to 2-connected Gorenstein curves, including singular cases, and characterizes the conditions for equality involving hyperelliptic and disconnected curves.

Findings

Established an inequality for h^0(C, I_S K_C) involving p_a(C) and deg(S).

Identified conditions under which equality holds, linking to hyperelliptic and disconnected curves.

Generalized classical Clifford's theorem to singular and reducible curves.

Abstract

Let C be a 2-connected Gorenstein curve either reduced or contained in a smooth algebraic surface and let S be a subcanonical cluster (i.e. a 0-dim scheme such that the space H^0(C, I_S K_C) contains a generically invertible section). Under some general assumptions on S or C we show that h^0(C, I_S K_C) <= p_a(C) - deg (S)/2 and if equality holds then either S is trivial, or C is honestly hyperelliptic or 3-disconnected. As a corollary we give a generalization of Clifford's theorem for reduced curves.