Let X be a smooth
projective variety, let L be a very ample invertible sheaf on X and assume
N + 1 =dim(H0(X,L)), the dimension of the space of global sections of L. Let
P1,…,Pt be general points on X and consider the blowing-up π : Y → X of X at
those points. Let Ei= π−1(Pi) be the exceptional divisors of this blowing-up.
Consider the invertible sheaf M := π∗(L) ⊗ OY(−E1−…− Et) on Y . In case
t ≤ N + 1, the space of global section H0(Y,M) has dimension N + 1 −t. In case this
dimension N + 1 −t is at least equal to 2dim(X) + 2, hence t ≤ N − 2dim(X) − 1, it
is natural to ask for conditions 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). For the projective plane
P2 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′⊗k
for some k ≥ 3dim(X) + 1 with L′ a very ample invertible sheaf on X: If
t ≤ N − 2dim(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 a very
ample invertible sheaf on X satisfying Cliff (L) ≥ 3 (“most” invertible sheaves
on X satisfy that property on the Clifford index), then M is very ample if
t ≤ N − 5. Examples show that the condition on the Clifford index cannot be
omitted.
