k-very ample line bundles on Del Pezzo Surfaces

TL;DR

This paper provides a numerical characterization of k-very ample line bundles on Del Pezzo Surfaces, emphasizing the role of exceptional curves and effective divisors in determining nefness and k-very ampleness.

Contribution

It offers a complete numerical description of the set of divisors critical for checking k-very ampleness on Del Pezzo Surfaces, linking geometric properties to intersection numbers.

Findings

Line bundle L is nef iff it is spanned.

L is k-very ample iff intersection with all elements of S is ≥ k.

Complete numerical description of the set S of divisors.

Abstract

A $k$-very ample line bundle L on a Del Pezzo Surface is numerically characterized. We find that the set of the exceptional curves and effective divisors with selfintersection zero, $\cal S$, plays a very important rule in checking the nefness and $k$-very ampleness of a line bundle on a Del pezzo Surface. We show that a line bundle L is nef if and only if it is spanned and that L is $k$-very ample if and only if the intersection with all the elements of $\cal S$ is greater or equal than k. In the first part of the paper we give a complete numerical description of the elements of $\cal S$ .