×

zbMATH — the first resource for mathematics

On Fano manifolds of large index. (English) Zbl 0726.14028
This paper is a continuation of the author’s investigation about the characterization of Fano manifolds [ibid. 68, No.2, 135-141 (1990; Zbl 0715.14033)], where Fano manifolds X with \(n\leq 2r-2\) were studied where \(n=\dim (X)\) and r its index.
In this note, Fano manifolds with \(n=2r-1\) are studied. The author shows that for such manifolds their Picard numbers is equal to 1 except in 3 cases. - The main tool of this paper is to use an idea of Mori to construct a family of deformations of a rational curve on X.

MSC:
14J45 Fano varieties
14C22 Picard groups
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [Katata] Birational Geometry of Algebraic Varieties. Open Problem List from The 23rd International Symposium of the Division of Mathematics of the Taniguchi Foundation held in Katata in August 1988
[2] [F] Fujita, T.: On adjoint bundles of ample vector bundles, preprint · Zbl 0782.14018
[3] [F1] Fujita, T.: On polarized manifolds whose adjoint bundle are not semipositive, in Algebraic Geometry, Sendai 1985, Adv. Studies in Pure Math., vol. 10, Kinokuniya and North Holland, 1987, 167–178
[4] [I] Ionescu, P.: Generalized Adjunction and Applications, Math. Proc. Cambridge Phil. Soc.99, 457–472 (1986) · Zbl 0619.14004 · doi:10.1017/S0305004100064409
[5] [KMM] Kawamata, Y., Matsuda, K. and Matsuki, K.: Introduction to the Minimal Model Problem. Adv. in Pure Math 10 (1987) · Zbl 0672.14006
[6] [M1] Mori, S.: Projective Manifolds with Ample Tangent Bundle, Ann. Math.,110, 593–606 (1979) · Zbl 0423.14006 · doi:10.2307/1971241
[7] [M2] Mori, S.: Cone of Curves and Fano Manifolds, in Proceedings of the International Congress of Mathematicians, Warszawa 1983, PWN/North-Holland 1984, 747–752
[8] [P] Peternel, T.: On ample vector bundles on Fano manifolds, preprint
[9] [S] Serre, J.-P.: Algébre Locale. Multiplicités, 1957/58, 3rd edition, Springer Lecture Notes in Math. 11, 1975
[10] [W] Wiśniewski J.A.: On a conjecture of Mukai, manuscripta math.68, 135–141 · Zbl 0715.14033
[11] [W1] Wiśniewski, J.A.: Length of Extremal Rays and Generalized Adjunction, Math. Zeit.200, 409–427 (1989) · Zbl 0668.14004 · doi:10.1007/BF01215656
[12] [W2] Wiśniewski, J.A.: On Contractions of Extremal Rays of Fano Manifolds, to appear in J. Reine Angew. Math. · Zbl 0721.14023
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.