On Clifford's theorem for singular curves

Let C be a 2-connected projective curve either reduced with planar singularities or contained in a smooth algebraic surface and let S be a subcanonical cluster (that is, a zero-dimensional scheme such that the space H0(C, ℐS KC) contains a generically invertible section). Under some general assumptions on S or C, we show that h0(C, ℐS KC)≤pa(C)−½ deg (S) 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 with planar singularities.