#### Vol. 9, No. 3, 2015

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Secant spaces and syzygies of special line bundles on curves

### Marian Aprodu and Edoardo Sernesi

Vol. 9 (2015), No. 3, 585–600
##### 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 $L={K}_{C}$.

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