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On the projectively almost-factorial varieties

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Summary

A projectively normal subvariety (X,O X) ofP N(k), k an algebraically closed field of characteristic zero, 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,O X) is projectively almost-factorial if and only if for all x ∈ X the local ringO x is almost-factorial and the quotient ofPic(X) modulo the subgroup generated by the class ofO X (1) is torsion. We also prove the invariance of the projective almost-factoriality up to isomorphisms and state some relations between the projective almost-factoriality (resp. projective factoriality) of X and the almost-factoriality (resp. factoriality) of the affine open subvarieties. Finally we discuss some consequences of the main result in the case k=ℂ: in particular we prove that the Picard group of a projectively almost-factorial variety is isomorphic to the Néron-Severi group, hence finitely generated.

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Entrata in Redazione il 23 aprile 1976.

AMS(MOS) subject classification (1970): Primary 14C20, 13F15.

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Beltrametti, M.C., Odetti, F.L. On the projectively almost-factorial varieties. Annali di Matematica 113, 255–263 (1977). https://doi.org/10.1007/BF02418376

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Keywords

  • Complete Intersection
  • Characteristic Zero
  • Picard Group
  • Projective Factoriality
  • Weil Divisor