#### 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 $L={K}_{C}$.

##### Keywords
projective curves, Brill–Noether theory, syzygies, secant loci
##### Mathematical Subject Classification 2010
Primary: 14N05
Secondary: 14N25, 14M12
##### Milestones
Revised: 27 January 2015
Accepted: 2 March 2015
Published: 17 April 2015
##### Authors
 Marian Aprodu Simion Stoilow Institute of Mathematics of the Romanian Academy P.O. Box 1-764 014700 Bucharest Romania Faculty of Mathematics and Computer Science University of Bucharest 14 Academiei Street 010014 Bucharest Romania Edoardo Sernesi Dipartimento di Matematica e Fisica Università degli Studi Roma Tre Largo San Leonardo Murialdo I-00146 Roma Italy