On ample vector bundles whose adjunction bundles are not numerically effective (Q921075)

By a generalized polarized variety the authors mean an algebraic variety X of dimension n over an algebraically closed field of characteristic zero together with an ample vector bundle E on it, i.e. a pair (X,E). Then \(K_ X+c_ 1(E)\) is its associated adjunction bundle. The paper studies the numerical effectiveness of this bundle. Most of the results from \textit{T. Fujita}'s paper [Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 167-178 (1987; Zbl 0659.14002)] about adjunction bundles of polarized varieties are generalized to those of generalized varieties. Main results: Theorem 1. If the rank of E is \(n+1\) then \(K_ X+c_ 1(E)\) is always numerically effective and \(K_ X+c_ 1(E)\) is equal to zero if and only if \((X,E)=(P^ n,\oplus^{n+1}{\mathcal O}_{P^ n}(1)).\) Theorem 2. If the rank of E is n then \(K_ X+c_ 1(E)\) is numerically effective unless \((X,E)=(P^ n,\oplus^ nO_ Pn(1)).\) Theorem 3. If the rank of E is \(n-1\) then \(K_ X+c_ 1(E)\) is numerically effective unless (X,E) is one of the following: 1. X is a scroll over a smooth curve and \(E|_ F=\oplus^{n-1}O_ Pn-1(1)\), where \(F=P^{n-1}\) is any fiber of this scroll. 2. \((P^ n,\oplus^{n-1}{\mathcal O}_ Pn(1))\). 3. \((P^ n,\oplus^{n- 2}{\mathcal O}_ Pn(1)\oplus {\mathcal O}_ P^ n(2))\). 4. \((Q^ n,\oplus^{n-1}{\mathcal O}_ Qn(1))\). Here \(Q^ n\) is a smooth hyperquadric. Theorem 4. If the rank of E is n and \(c_ 1(E)=c_ 1(X)\), then P(E) is a Fano-\((2n-2)\)-fold with index n and Picard number two. Moreover P(E) has exactly two extremal rays, and the contraction maps associated with these extremal rays are both of fiber type and equidimensional unless \((X,E)=(P^ n,\oplus^{n-1}O_ Pn(1)\oplus O_ Pn(2))\) or \((Q^ 2,O_ Q2(1)\oplus O_ Q2(1))\).