Explicit presentations of nonspecial line bundles and secant spaces

TL;DR

This paper studies explicit minimal presentations of nonspecial line bundles on smooth curves, exploring conditions for very ampleness and secant spaces, and applies findings to verify conjectures on property (N_p) for certain curves.

Contribution

It introduces the concept of minimal presentations of nonspecial line bundles and investigates their properties and applications to secant spaces and conjectures.

Findings

Minimal presentations lead to very ample line bundles with specific secant space properties.

Constructs examples of nonspecial line bundles that test Green and Lazarsfeld's Conjecture.

Provides conditions under which divisors D and E produce minimal presentations with desired geometric features.

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.