Beltrametti, Mauro C.; Sommese, Andrew J. On the adjunction theoretic classification of polarized varieties. (English) Zbl 0762.14005 J. Reine Angew. Math. 427, 157-192 (1992). The paper under review presents some of the recent results on the adjunction classification of polarized algebraic varieties. If \({\mathcal L}\) is an ample line bundle on a smooth projective algebraic variety \({\mathcal M}\) of dimension \(n\), then except of a short list of exceptions [cf. T. Fujita in Algebraic geometry, Proc. Symp., Sendai 1985, Adv. Stud. Pure Math. 10, 167-178 (1987; Zbl 0659.14002); P. Ionescu, Math. Proc. Camb. Philos. Soc. 99, 457-472 (1986; Zbl 0619.14004) and A. J. Sommese in Complex Analysis and Algebraic Geometry, Proc. Conf., Göttingen 1985, Lect. Notes Math. 1194, 175-213 (1986; Zbl 0601.14029)], the existence of the second reduction \((X,{\mathcal K})\) of \(({\mathcal M},{\mathcal L})\) is known. The main result states that for \(n\geq 6\), \(K_ X+(n-3){\mathcal K}\) is nef unless \(n=6\) and either (a) \((X,{\mathcal K})\cong(\mathbb{P}^ 6,{\mathcal O}(1))\), or (b) \(K_ X=-5{\mathcal K}\), and \((X,{\mathcal K})\) is a Gorenstein Del Pezzo variety. There is also a partial result, theorem 5.3, for 5-folds. Reviewer: V.Shokurov (Moskva) Cited in 3 ReviewsCited in 17 Documents MSC: 14C20 Divisors, linear systems, invertible sheaves 14E05 Rational and birational maps 14J40 \(n\)-folds (\(n>4\)) 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli Keywords:adjunction classification of polarized algebra varieties; Del Pezzo variety; 5-folds PDF BibTeX XML Cite \textit{M. C. Beltrametti} and \textit{A. J. Sommese}, J. Reine Angew. Math. 427, 157--192 (1992; Zbl 0762.14005) Full Text: DOI Crelle EuDML