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=K_X -D +E for some effective divisors D and E>0 on X with gcd (D, E)=0 and h^0 (X, O_X (D))=1. In this work, we define a minimal presentation of L which is minimal with respect to the degree of E among the presentations. If L=K_X -D +E with degE>2 is a minimal, then L is very ample and any q-points of X with q <degE are embedded in general position but the points of E are not. We investigate sufficient conditions on divisors D and E for L=K_X -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=K_X -D +E on X with/without an n-secant (n-2)-plane of the embedded curve by taking divisors D and E on X. As its applications, we construct nonspecial line bundles which show the sharpness of Green and Lazarsfeld's Conjecture on property (N_p) for general n-gonal curves and simple multiple coverings of smooth plane curves.