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A criterion for a rational projectively normal variety to be almost- factorial. (English) Zbl 0731.14028

Let \(F\hookrightarrow {\mathbb{P}}^ N\) be a rational, projectively normal variety, \(\dim (F)=n\). The notion of a parametrization p on \({\mathbb{P}}^ N\) for the variety F and of subvarieties of F referred to p is introduced. Under these assumptions, the following criterion for F to be almost-factorial is proved which is a generalization of the well-known Gallarati criterion for monoid hypersurfaces to be almost-factorial:
Theorem. The following conditions are equivalent: (a) F is almost- factorial; (b) every simple divisor on F is a set-theoretic complete intersection on F.
Examples of rational, almost-factorial varieties are presented which are isomorphic neither to a projective space nor to a monoid. Besides, the question of classification of almost-factorial rational surfaces of degree \( 4\) having only ordinary double points as singularities is discussed.

MSC:

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14M20 Rational and unirational varieties
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References:

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