Secant spaces and syzygies of special line bundles on curves

On a special line bundle L on a projective curve C we introduce a geometric condition called (Δ<inf>q</inf>). When L D KC, this condition implies gon(C) ≥ q +2. For an arbitrary special L, we show that (Δ<inf>3</inf>) implies that L has the well-known property (M<inf>3</inf>), generalising a similar result proved by Voisin in the case L = K<inf>C</inf>.