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Une base de \(Pic(Hilb^ k{\mathbb{P}}^ 2)\). (French) Zbl 0548.14003

By work of Iarrobino and Fogarty one knows that \(Pic(Hilb^ k{\mathbb{P}}^ 2)={\mathbb{Z}}^ 2\). In this paper an explicit basis is given for this group: it consists of the divisor of all k-uples meeting a fixed line and the divisor of all k-uples containing a double point which is colinear with a fixed point. As an application it is proved that \(Pic(Hilb^ 3{\mathbb{P}}^ N)={\mathbb{Z}}^ 2\) and a description of a basis for it is also given: it consists of the divisor of all triples meeting a fixed hyperplane in \({\mathbb{P}}^ N\) and the divisor of all triples whose plane meets a fixed codimension 3 linear subspace of \({\mathbb{P}}^ N\).
Reviewer: A.Buium

MSC:

14C22 Picard groups
14C05 Parametrization (Chow and Hilbert schemes)
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