Abh. Math. Sem. Univ. Hamburg 63 (1993), 215-226 The k-very Ampleness on an Elliptic Quasi Bundle By M. MELLA and M. PALLESCHI Introduction BELTRAMETTIand SOMMESEhave recently introduced the definition of a k-very ample line bundle in connection with the notion of a high order embedding, [4]. More precisely, let S be a smooth projective surface over II~. A line bundle L on S is k-very ample (k > 0) if for any 0-dimensional subscheme (Z, (gz) of S with k + 1 = length((fz), the restriction map r(L) , r((~z(L)) is a surjection. Note that the 0-very ampleness of L is equivalent to L being spanned by its global sections and that the 1-very ampleness of it is the same property as its very ampleness. For higher values of k the k-very ampleness of L gives information on the model S' obtained by embedding S into IP N via the morphism associated with L. Roughly speaking, S' turns out not to contain k + 1 points belonging to the same ]pk-1 of pN. We finally recall that, for k < 2, this notion is equivalent to the k-spannedness introduced in [2]. By taking advantage of a Reider-type criterion proved by BELa-RAMETTI and SOMMESZ([4]), it is possible to work out numerical criteria for the k-very ampleness of a line bundle on a surface S whose Num(S) is fully described. This has already been done in the case when S is a Fl-bundle over a curve ([3]) and for its blowing-ups ([1]). In this paper S is a relatively minimal elliptic fibration over a rational curve, with b2 = 2 and Kodaira dimension x(S) > O. First of all, all the line bundles on S spanned by their global sections are described (Proposition 2.4). Secondly we supply both necessary condi- tions (Propositions 2.7 and 2.9) and sufficient ones (Propsitions 2.5 and 2.8, Remark 2.6) for a line bundle to be k-very ample. These results become far more precise in the case when S is a hyperel- liptic surface. Such surfaces were classified by BAGNERA and DE FRANCHIS, who divided them into 7 types. For each one of them we give a complete characterization of the k-very ampleness of a line bundle on S. In order to study the k-very ampleness on an elliptic fibration we have had to analyze the k-very ampleness of a line bundle on a smooth curve C (the definition is formally the same as that for a surface). In this case we characterized k-gonal curves C in terms of the (k- 1)-very ampleness of their canonical bundle (Proposition 1.2) in this way generalizing a classical fact