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Picard numbers of surfaces in 3-dimensional weighted projective spaces. (English) Zbl 0695.14016
Let \(q_ 0,...,q_ 3\) be positive integers such that each triple among them has gcd equal to 1. Let \({\mathbb{P}}={\mathbb{P}}(q_ 0,...,q_ 3)\) be the associated weighted projective space. Then Pic(\({\mathbb{P}})\cong {\mathbb{Z}}\). Let \({\mathcal L}\) be an ample generator of Pic(\({\mathbb{P}})\). The authors give sufficient conditions (in terms of the weights) in order that a general element of the linear system \(| {\mathcal L}|\) on \({\mathbb{P}}\) is a surface with Picard number equal to 1, thus supplementing the result of D. A. Cox [Math. Z. 201, No.2, 183-189 (1989; Zbl 0686.14041)] who showed that a general element of \(| {\mathcal L}^{\otimes 2}|\) either has \(p_ g=0\) or Picard number equal to 1.
Reviewer: A.J.de Jong

14C22 Picard groups
14C20 Divisors, linear systems, invertible sheaves
Zbl 0686.14041
Full Text: DOI EuDML
[1] Carlson, J., Green, M., Griffiths, P., Harris, J.: Infinitesimal variation of Hodge structure I Compos. Math.50, 109–205 (1983) · Zbl 0531.14006
[2] Cox, D.: Picard numbers of surfaces in 3-dimensional weighted projective spaces. Math. Z.201, 183–189 (1989) · Zbl 0686.14041
[3] Delorme, Ch.: Espaces projectifs anisotropes. Bull. Soc. Math. Fr.103, 203–223 (1975) · Zbl 0314.14016
[4] Shioda, T.: The Hodge conjecture for Fermat varieties. Math. Ann.245, 175–184 (1979) · Zbl 0408.14012
[5] Steenbrink, J.H.M.: Intersection form for quasi-homogeneous singularities. Compos. Math.34, 221–223 (1977) · Zbl 0347.14001
[6] Steenbrink, J.H.M.: On the Picard group of certain smooth surfaces in weighted projective spaces. In: Aroca, J.M., Buchweitz, R., Giusti, M., Merle, M. (ed.) Algebraic Geometry. Proceedings, La Rábida 1981. (Lecture Notes Math. vol. 961, pp. 302–313) Berlin Heidelberg New York, Springer 1982
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