an:00013435
Zbl 0754.14027
Beltrametti, Mauro C.; Sommese, Andrew J.
New properties of special varieties arising from adjunction theory
EN
J. Math. Soc. Japan 43, No. 2, 381-412 (1991).
00157535
1991
j
14J60 14C20 14M99 14D05
Del Pezzo manifolds; extremal ray; very ample line bundle; adjunction; Mori theory
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. \textit{M. Beltrametti}, \textit{A. J. Sommese} and \textit{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); \textit{M. Andreatta}, \textit{E. Ballico} and \textit{J. Wisniewski}, Int. J. Math. 3, No. 3, 331-340 (1992) and \textit{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 \textit{J. Koll??r} [``Effective base point freeness'' (preprint), see e.g. \textit{M. Andreatta}, \textit{E. Ballico} and \textit{J. Wisniewski}, ``Two theorems on elementary contractions''].
E.Ballico (Povo)