Secant spaces and syzygies of special line bundles on curves

Abstract

On a special line bundle $L$ on a projective curve $C$ we introduce a geometric condition called $\left({\Delta }_q\right)$. When $L=K_C$, this condition implies $gon\left(C\right)\ge q+2$. For an arbitrary special $L$, we show that $\left({\Delta }_3\right)$ implies that $L$ has the well-known property $\left(M_3\right)$, generalising a similar result proved by Voisin in the case $L=K_C$.