Secant planes of projective curves embedded by nonspecial line bundles

For a nonspecial line bundle ? on a smooth curve X we consider a presentation ??𝒦_X?D+E which is minimal with respect to deg E. If ? is very ample, then this minimality means that any n-points of φ_?(X) with n≤deg E?1 are in general position while φ_?(E) spans a (deg E?2)-plane. In this work, we investigate conditions on D and E for ??𝒦_X?D+E to be minimal. We also observe s-secant (s?k?1)-planes which are minimal with respect to the secant degree s for a given k. We apply minimal presentations to problems about the exactness of Green-Lazarsfeld’s conjecture on property (N_p).