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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

MSC:
14C22 Picard groups
14C20 Divisors, linear systems, invertible sheaves
Citations:
Zbl 0686.14041
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References:
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