  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 $\cal S$ .
