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On the adjunction theoretic classification of polarized varieties. (English) Zbl 0762.14005
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.

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
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