| Abstract: | 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.
Entrata in Redazione il 23 aprile 1976.
AMS(MOS) subject classification (1970): Primary 14C20, 13F15. |
