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On the projectively almost-factorial varieties. (English) Zbl 0358.14025

From authors’ abstract: A projectively normal subvariety \((X,{\mathcal O}_X)\) of \(\mathbb P^N(k)\), \(k\) an algebraically closed field of characteristic \(0\), will be said to be projectively almost-factorial if each Weil divisor has a multiple which is a complete intersection in \(X\). The main result is the following: \((X,{\mathcal O}_X)\) is projectively almost-factorial if and only if for all \(x \in X\) the local ring \({\mathcal O}_X\) is almost-factorial and the quotient of Pic\((X)\) modulo the subgroup generated by the class of \({\mathcal O}_X(1)\) is torsion. A corollary: in the case \(k=\mathbb C\), the Picard group of a projectively almost-factorial variety is isomorphic to the Néron-Severi group, hence finitely generated.
Reviewer: L. N. Vaserstein

MSC:

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14C20 Divisors, linear systems, invertible sheaves
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