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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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