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Very ample linear systems on blowings-up at general points of smooth projective varieties. (English) Zbl 1059.14011

Author’s abstract: Let \(X\) be a smooth projective variety, let \(L\) be a very ample sheaf on \(X\) and assume \(N + 1 = \dim(H^{0}(X,L))\), the dimension of the space of global sections on \(L\). Let \(P_1, \ldots ,P_t\) be general points on \(X\) and consider the blowing up \(\pi : Y \rightarrow X\) of \(X\) at those points. Let \(E_i = \pi^{-1}(P_i)\) (\(i = 1, \ldots , t\)) be the exceptional divisors of this blowing up. Consider the invertible sheaf \(M := \pi^*(L) \otimes O_Y(-E_1 - \cdots - E_t)\) on \(Y\). In case \(t \leq N + 1\), the space of global sections \(H^0(Y,M)\) has dimension \(N + 1 - t\). In case this dimension \(N + 1 - t\) is at least equal to \(2 \dim(X) + 2\), hence \(t \leq N - 2 \dim(X) - 1\), 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 \(2 n \)). For the projective plane \(\mathbb{P}^2\) this problem was solved by J. d’Almeida and A. Hirschowitz [Math. Z. 211, 479–483 (1992; Zbl 0759.14004)].
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 \(L = L^{\prime\; \otimes k}\) for some \(k \geq 3 \dim(X) + 1\) with \(L'\) a very ample invertible sheaf on \(X\): if \(t \leq N - 2 \dim(X) - 1\) then \(M\) is very ample on \(Y\). Using the same method of proof, we obtain a very sharp result when \(X\) is a \(K3\)-surface: let \(L\) be a very ample invertible sheaf on \(X\) satisfying \(\text{Cliff}(L) \geq 3\) (“most” invertible sheaves on \(X\) satisfy that property on the Clifford index), then \(M\) is very ample if \(t \leq N - 5\). Examples show that the condition on the Clifford index cannot be omitted.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14N05 Projective techniques in algebraic geometry
14N25 Varieties of low degree

Citations:

Zbl 0759.14004
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