Very ample linear systems on blowings-up at general points of smooth projective varieties

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 ,…, Pt be general points on X and consider the blowing-up π: Y → X of X at those points. Let E i = π -1 (P i ) be the exceptional divisors of this blowing-up. Consider the invertible sheaf M:= π * (L) O Y (-E 1 - … - E t ) on Y. In case t 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.