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Ample divisors on the blow up of \(\mathbb{P}^3\) at points. (English) Zbl 0906.14003

Let \(n=2\) or \(3\), and \(\mathbb{P}^n\) be the projective space over \(\mathbb{C}\). Let \(p_1,\ldots,p_k\) be \(k\) points in \(\mathbb{P}^n\) in general position and \(\pi\: X\rightarrow \mathbb{P}^n\) be the blow up of \(\mathbb{P}^n\) at \(p_1,\ldots,p_k\) with exceptional divisors \(E_1,\ldots,E_k\). Let \(H=\pi^*\mathcal O_{\mathbb{P}^n}(1)\). The author proves that, if \(d\geq d_0(n)\), the divisor \(L=dH-\sum_{i=1}^k E_i\) is ample if and only if \(L^n>0\), i.e., \(d^n>k\), where \(d_0(2)=3\), \(d_0(3)=5\). This results extends a theorem of G. Xu [Manuscr. Math. 86, No. 2, 195-197 (1995; Zbl 0836.14004)] on the blow up of \(\mathbb{P}^2\).
Reviewer: V.L.Popov (Moskva)

MSC:

14C20 Divisors, linear systems, invertible sheaves
14N05 Projective techniques in algebraic geometry

Citations:

Zbl 0836.14004
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References:

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