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

Beltrametti, Mauro C.; Sommese, Andrew J.

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)

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