Clifford index for reduced curves

TL;DR

This paper extends the Clifford index concept to reduced curves with singularities using torsion free sheaves, analyzing its behavior and verifying Green's conjecture for specific reducible curves.

Contribution

It introduces a new approach to Clifford index for singular curves and proves Green's conjecture for certain reducible cases.

Findings

Clifford index extended to singular curves using torsion free sheaves.

Green's conjecture verified for some reducible curves.

Behavior of Clifford index linked to curve's combinatorial properties.

Abstract

We extend the notion of Clifford index to reduced curves with planar singularities by considering rank 1 torsion free sheaves. We investigate the behaviour of the Clifford index with respect to the combinatorial properties of the curve and we show that Green's conjecture holds for certain classes of curves given by the union of two irreducible components.