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Projective n-folds of log-general type. I. (English) Zbl 0702.14037
In this paper adjunction theory is used to study the projective classification of a normal Gorenstein n-dimensional projective variety, X, which has singularities of codimension \(\geq 3\), whose irrational singularities are isolated, and which is embedded in \({\mathbb{P}}_{{\mathbb{C}}}\) by the sections of a very ample line bundle, L, which satisfies \(h^ 0((K_ X\otimes (n-2)L)^ N)\neq 0\) for some \(N>0\). The results in the paper are used to study the classification of 3-folds with the genus of a curve section \(\leq 14\). This list has been used to classify 3-folds in \({\mathbb{P}}^ 5\) of degree 9, 10, and 11 [see M. Beltrametti, M. Schneider and A. Sommese “The threefolds of degree 9 and 10 in \({\mathbb{P}}^ 5\)” (to appear in Math. Ann.) and “The threefolds of degree 11 in \({\mathbb{P}}^ 5\)” (to appear in 1989 Bergen Proc., Lect. Notes. Math.)]
Further developments: M. Beltrametti, M. Fania, and A. Sommese, “On projective classification of algebraic varieties via adjunction theory” (preprint) and A. Sommese, J. Reine Angew. Math. 402, 211-220 (1989; Zbl 0675.14005).

MSC:
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14N05 Projective techniques in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
14J30 \(3\)-folds
14J10 Families, moduli, classification: algebraic theory
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