Angelini, Flavio Ample divisors on the blow up of \(\mathbb{P}^3\) at points. (English) Zbl 0906.14003 Manuscr. Math. 93, No. 1, 39-48 (1997). 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) Cited in 1 ReviewCited in 3 Documents MathOverflow Questions: Ample divisors on blown-up projective space MSC: 14C20 Divisors, linear systems, invertible sheaves 14N05 Projective techniques in algebraic geometry Keywords:blow up; projective space; Grassmannian; linear system; exceptional divisors Citations:Zbl 0836.14004 PDFBibTeX XMLCite \textit{F. Angelini}, Manuscr. Math. 93, No. 1, 39--48 (1997; Zbl 0906.14003) Full Text: DOI arXiv EuDML References: [1] F. Angelini,UCLA Thesis, (1995). [2] L. Ein, R. Lazarsfeld,Seshadri constants on smooth surfaces, S. M. F. Astérisque,218, (1993), 177–186. · Zbl 0812.14027 [3] M. Green,A new proof of the explicit Noether-Lefschetz theorem, J. Differential Geom.,27, (1988), 155–159. · Zbl 0674.14005 [4] P. Griffiths, J. Harris,Principles of algebraic geometry, Wiley Interscience, New York, 1978. · Zbl 0408.14001 [5] P. Griffiths, J. Harris,On the Noether-Lefschetz theorem and some remarks on codimension two cycles, Math. Ann.,271, (1985), 31–51. · Zbl 0552.14011 · doi:10.1007/BF01455794 [6] R. Hartshorne,Algebraic geometry, Springer-Verlag, New York, 1977. · Zbl 0367.14001 [7] R. Lazarsfeld,Lecture on Linear Series, (to appear). · Zbl 0906.14002 [8] S. Kim,Noether-Lefschetz locus for surfaces, Trans. Amer. Math. Soc.,324, (1991), 369–384. · Zbl 0739.14019 · doi:10.2307/2001513 [9] O. Küchle,Ample line bundles on blown up surfaces, Math. Ann.304, (1996), 151–155. · Zbl 0834.14024 · doi:10.1007/BF01446289 [10] G. Xu,Divisors on the blow up of the projective plane, Man. Math.86, (1995), 195–198. · Zbl 0836.14004 · doi:10.1007/BF02567988 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.