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.
The gonality and the Clifford index of curves on a toric surface
Normal generation and Clifford index
Gonality and Clifford Index of curves on $K3$ Surfaces
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