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New properties of special varieties arising from adjunction theory. (English) Zbl 0754.14027
Fix a pair $$(X,L)$$ with $$X$$ an $$n$$-dimensional complex projective manifold and $$L$$ a very ample line bundle on $$X$$. This paper (as well much research) is concerned with the geometry connected with the adjunction maps (say $$\pi:X\to Y)$$ given by $$| K_ X+rL|$$ and $$| t(K+rL)|$$. This paper gives strong informations in 3 critical cases: for $$r=n-2$$ quadric bundles over surfaces and Del Pezzo fibrations over curves and for $$n=3$$, $$t=2$$, $$2r=3$$ fibrations over curves with $$(\mathbb{P}^ 2,\mathbb{O}(2))$$ as general fiber. A strong motivation came from projective geometry: e.g. the Del Pezzo part was applied elsewhere to the classification of 3-folds of degree 9 and 10 in $$\mathbb{P}^ 5$$. A main result is that if $$Y$$ is a normal surface, then $$Y$$ has at most $$A_ 1$$ singularities and if $$n\geq 4$$, then $$\pi$$ is equidimensional.
Later, large parts of the paper were generalized and the conjectures raised here partly solved [see e.g. M. Beltrametti, A. J. Sommese and J. Wisniewski: “Results on varieties with many lines and their applications to adjunction theory” in: Complex algebraic varieties, Proc. Conf., Bayreuth 1990, Lect. Notes Math. 1507, 33-38 (1992); M. Andreatta, E. Ballico and J. Wisniewski, Int. J. Math. 3, No. 3, 331-340 (1992) and G. Besana, “On the geometry of conic bundles arising in adjunction theory” Ph. D. thesis Notre Dame 1992].
In recent papers (by Mori theory) the interest was mainly in the case “$$L$$ ample”. An extremely strong tool for this case is the improved version of the proof of Kawamata’s base point free theorem given by J. Kollár [“Effective base point freeness” (preprint), see e.g. M. Andreatta, E. Ballico and J. Wisniewski, “Two theorems on elementary contractions”].
Reviewer: E.Ballico (Povo)

##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14C20 Divisors, linear systems, invertible sheaves 14M99 Special varieties 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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