On Reider’s method and higher order embeddings

k—Very Ample Line Bundles on Del Pezzo Surfaces

“…Therefore any line bundle on S r is of the form al − r 1 b i e i and from the definition of blow up it follows that −K S = 3l − r 1 e i . Since for S 0 and for S 1 = P 1 × P 1 a complete characterization of k-very ample line bundles has been done in [2] and [5], from now on we'll focus our attention on S r = B P1,...,Pr for r = 1, ..., 8, as in proposition 2.4 d). We will use the following notation, according to [8]: Let I r = {L ∈ P ic(S r ) such that L 2 = −1 and LK S = −1} be the set of exceptional curves on S r , they are in fact irreducible effective divisors.…”

“…Using the Reider-type criterion (theorem 2.3) it is possible to give an exact numerical characterization of k-very ample line bundles on Surfaces whose Picard group is fully known. k-very ample line bundles on P 2 , on the Hirzebruch surfaces F n and on hyperelliptic surfaces are indeed completely characterized in [2], [5], [10].…”

On k-Jet Ampleness of Line Bundles on Hyperelliptic Surfaces

“…The concepts of higher order embeddings: k-spandness, k-very ampleness, and k-jet ampleness were introduced and studied in a series of papers by Beltrametti, Francia, and Sommese, see [6][7][8]. The last notion is of our main interest in the present work.…”

A Characterization of the Veronese Surface

“…if for all points P e Supp(Z), Z is given around P by equations x = ym = 0, x and y suitable local coordinates around P. According to is called Âc-spanned if the linear system on S determined by the embedding in P" /c-spans 5 . There are other, perhaps more natural, definitions of /c-spannedness (see [BFS, §4]), but not only is the one given here is the one used heavily in [BFS] and [BS], but also it seems the weakest one among the natural possible definitions, and so the one with which …”

“…A smooth surface 5cP is called Âc-spanned if the linear system on S determined by the embedding in P" /c-spans 5 . There are other, perhaps more natural, definitions of /c-spannedness (see [BFS,§4]), but not only is the one given here is the one used heavily in [BFS] and [BS], but also it seems the weakest one among the natural possible definitions, and so the one with which Theorem 1 is strongest. It is easy to check that if W is (k + l)-spanned, then it is A:-spanned.…”