Explicit presentations of nonspecial line bundles and secant spaces

Abstract

A line bundle L on a smooth curve X is nonspecial if and only if L admits a presentation L ≃ KX − D + E for some divisors D ≥ 0, E > 0 on X with gcd(D, E) = 0 and h 0 (X, OX (D)) = 1. In this work, we define a minimal presentation of L which is minimal with respect to degE among the presentations. If L ≃ KX − D + E with degE ≥ 3 is a minimal, then L is very ample and any q-points of ϕL(X) with q ≤ degE − 1 are in general position but the points of ϕL(E) are not. We investigate sufficient conditions on divisors D, E for L ≃ KX − D + E to be minimal. Through this, for a number n in some range, it is possible to construct a nonspecial very ample line bundle L ≃ KX − D + E on X with/without an n-secant (n − 2)-plane of the embedded curve by taking divisors D, E on X. As its applications, we construct nonspecial line bundles which show the sharpness of Green and Lazarsfeld's Conjecture on property (Np) for general n-gonal curves and simple multiple coverings of smooth plane curves .