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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.