Pacific Journal of Mathematics vol:202 issue:2 pages:313-327

Abstract:

Let X be a smooth projective variety, let L be a very ample invertible sheaf on X and assume N + 1 = dim( H 0 ( X, L)), the dimension of the space of global sections of L. Let P-1,...,P-t be general points on X and consider the blowing-up pi : Y --> X of X at those points. Let E-i = pi(-1) (P-i) be the exceptional divisors of this blowing-up. Consider the invertible sheaf M : = pi*(L) circle times O-Y (-E-1...E-t) on Y. In case t less than or equal to N + 1, the space of global section H-0 (Y, M) has dimension N + 1-t. In case this dimension N + 1 is at least equal to 2 dim( X) + 2, hence t less than or equal to N-2 dim( X)-1, it is natural to ask forconditions implying M is very ample on Y ( this bound comes from the fact that "most" smooth varieties of dimension n cannot be embedded in a projective space of dimension at most 2n). Forthe projective plane P-2 this problem is solved by J. d'Almeida and A. Hirschowitz. The main theorem of this paper is a generalization of their result to the case of arbitrary smooth projective varieties under the following condition. Assume L = L'(circle timesk) forsome k greater than or equal to 3 dim( X) + 1 with L' a very ample invertible sheaf on X : If t less than or equal to N-2 dim(X)-1 then M is very ample on Y. Using the same method of proof we obtain very sharp result for K3-surface and let L be avery ample invertible sheaf on X satisfying Cliff (L) greater than or equal to 3 ("most" invertible sheaves on X satisfy that property on the Cli ord index), then M is very ample if t less than or equal to N-5. Examples show that the condition on the Clifford index cannot be omitted.