Very Ample Linear Systems on Blowings-up at General Points of Smooth Projective Varieties

Let X be a smooth projective variety, let L be a very ample invertible sheaf on X and assume N+1 = dim(H(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 H(Y,M) has dimension N + 1 − t. In case this dimension N + 1 − t is at least equal to 2 dim(X) + 2, hence t ≤ N − 2 dim(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 P 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 ≥ 3 dim(X) + 1 with L′ a very ample invertible sheaf on X: If t ≤ 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 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.