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 $({\mathrm{\Delta }}_q)$. When $L=K_C$ this condition implies gon$(C)\ge q+2$. For an arbitrary special $L$ we show that $({\mathrm{\Delta }}_3)$ implies that $L$ has the well-known property $(M_3)$, generalizing a similar result proved by Voisin in the case $L=K_C$.