New properties of special varieties arising from adjunction theory.

*(English)*Zbl 0754.14027Fix 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. KollĂˇr [“Effective base point freeness” (preprint), see e.g. M. Andreatta, E. Ballico and J. Wisniewski, “Two theorems on elementary contractions”].

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. KollĂˇr [“Effective base point freeness” (preprint), see e.g. M. Andreatta, E. Ballico and J. Wisniewski, “Two theorems on elementary contractions”].

Reviewer: E.Ballico (Povo)