zbMATH — the first resource for mathematics

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.

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
Full Text: DOI Crelle EuDML