Secant spaces and syzygies of special line bundles on curves

This is a joint work with Edoardo Sernesi. We study syzygies of curves embedded by special linear systems in connection with the varieties of secant planes. It is known that the existence of (p+2)-secant p-planes for small p represent an obstruction for vanishing of syzygies - for example, if the curve is embedded with a 3-secant line, then the ideal cannot be generated by quadrics. On the other hand, (p+2)-secant p-planes always exist for larger p. We show in some cases that a nice geometric behavior of the variety of secants, seen as a subvariety in the symmetric product, is a reason for vanishing of syzygies.