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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
Zbl 0686.14041
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##### References:
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