Beltrametti, Mauro C.; Schneider, Michael; Sommese, Andrew J. Applications of the Ein-Lazarsfeld criterion for spannedness of adjoint bundles. (English) Zbl 0804.14006 Math. Z. 214, No. 4, 593-599 (1993). In this paper using Ein-Lazarsfeld’s criteria for the spannedness of the adjoint bundle \(K_ X \otimes L\), of a nef and big line bundle, \(L\), on a complex projective 3-fold \(X\) and some inequalities for the Chern classes of ample and spanned vector bundles given by the authors in §1, they prove:If \(E\) is an ample and spanned vector bundle on a smooth 3-fold \(X\) then \(K_ X \otimes \text{det} (E)\) is spanned if \(\text{rank} (E) \geq 3\) and \(L^ 3 \geq 850\) or if \(\text{rank} (E) \geq 4\) and \(L^ 3 \geq 162\) or if \(\text{rank} (E) \geq 13\).In the last section, using again Ein-Lazarsfeld’s criteria the authors study projective \(n\)-folds without rational curves. Reviewer: R.M.Miró-Roig (Barcelona) Cited in 1 ReviewCited in 8 Documents MSC: 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14C20 Divisors, linear systems, invertible sheaves 14J30 \(3\)-folds 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli Keywords:\(n\)-folds without rational curves; spannedness of the adjoint bundle PDFBibTeX XMLCite \textit{M. C. Beltrametti} et al., Math. Z. 214, No. 4, 593--599 (1993; Zbl 0804.14006) Full Text: DOI EuDML References: [1] Ein, L., Lazarsfeld, R.: Global generation of pluricanonical and adjoint linear series on smooth projective threefolds. Preprint 1992 · Zbl 0803.14004 [2] Fujita, T.: Theorems of Bertini type for certain types of polarized manifolds. J. Math. Soc. Japan34, 709–718 (1982) · Zbl 0506.14005 [3] Fujita, T.: On adjoint bundles of ample vector bundles. In: Complex Algebraic Varieties, Bayreuth 1990. (Lect. Notes Math., vol. 1507, pp. 105–112) Berlin Heidelberg New York: Springer 1992 · Zbl 0782.14018 [4] Fulton, W.: Intersection Theory. (Ergeb. Math., Bd. 2) Berlin Heidelberg New York: Springer 1984 · Zbl 0541.14005 [5] Hartshorne, R.: Ample vector bundles on curves. Nagoya Math. J.43, 73–89 (1971) · Zbl 0218.14018 [6] Lanteri, A., Maeda, H.: Adjoint bundles of ample and spanned vector bundles on algebraic surfaces. J. reine angew. Math.433, 181–199 (1992) · Zbl 0753.14011 [7] Lanteri, A., Sommese, A.J.: A vector bundle characterization of \(\mathbb{P}\) n . Abh. Math. Semin. Univ. Hamburg58, 89–96 (1988) · Zbl 0701.14050 [8] Lazarsfeld, R.: A Barth-type theorem for branched coverings of projective space. Math. Ann.249, 153–162 (1980) · Zbl 0457.32006 [9] Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus 11. In: Raynaud, M., Shioda, T. (eds.) Algebraic Geometry, Japan-France Conference 1982. (Lect. Notes Math., vol. 1016, pp. 334–353) Berlin Heidelberg New York: Springer 1983 [10] Mumford, D.: Projective invariants of projective structures and applications. In: Proc. Int. Cong. Math. in Stockholm 1962, pp. 526–530. Djursholm: Institute Mittag-Leffler 1963 [11] Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math.82, 540–567 (1965) · Zbl 0171.04803 [12] Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math.127, 309–316 (1988) · Zbl 0663.14010 [13] Smyth, B., Sommese, A.J.: On the degree of the Gauss mapping of a submanifold of an Abelian variety. Comment. Math. Helv.59, 341–346 (1984) · Zbl 0549.14009 [14] Wiśniewski, J.A.: Length of extremal rays and generalized adjunction. Math. Z.200, 409–427 (1989) · Zbl 0668.14004 [15] Ye, Y.-G., Zhang, Q.: On ample vector bundles whose adjunction bundles are not numerically effective. Duke Math. J.60, 671–687 (1990) · Zbl 0709.14011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.