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Pseudo-Index of Fano Manifolds and Smooth Blow-Ups

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Abstract

Suppose π: X → Y is a smooth blow-up along a submanifold Z of Y between complex Fano manifolds X and Y of pseudo-indices i X and i Y respectively (recall that i X is defined by i X :=min {−K X ·C | C is a rational curve of X}). We prove that \(i_X \leq i_Y\) if 2 dim (Z) < dim (Y)+i Y −1 and show that this result is optimal by classifying the ‘boundary’ cases. As expected, these results are obtained by studying rational curves on X and Y.

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References

  1. Lazarsfeld, R.: Some applications of the theory of positive vector bundles,Complete Intersections,Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lecture Notes in Math. 1092, Springer, New York, 1984, pp 29–61

  2. V. A. Iskovskikh Yu. G. Prokhorov (1999) Fano Varieties Algebraic Geometry Springer-Verlag Berlin

    Google Scholar 

  3. Cho K., Miyaoka Y. and Shepherd-Barron, N.: Characterizations of projective space and applications to complex symplectic manifolds,Higher Dimensional Birational Geometry (Kyoto, 1997), Adv. Stud. Pure Math. 35, Math. Soc. Japan, Tokyo, 2002, pp. 1–88.

  4. L. Bonavero F. Campana et J. A. Wiśniewski (2002) ArticleTitleVariétés complexes dont l’éclatée en un point est de Fano C.R. Math. Acad. Sci. Paris 334 IssueID6 463–468

    Google Scholar 

  5. O. BonaveroL.CasagrandeC. Debarre S. Druel (2003) ArticleTitleSur une conjecture de Mukai Comment. Math. Helv 78 601–626 Occurrence Handle10.1007/s00014-003-0765-x

    Article  Google Scholar 

  6. Debarre O.: Higher-Dimensional Algebraic Geometry, Universitext, Springer Verlag, 2001.

  7. J. Kollár (1966) Rational Curves on Algebraic Varieties Springer-Verlag Berlin

    Google Scholar 

  8. C. Okonek M. Schneider H. Spindler (1980) Vector Bundles on Complex Projective Spaces Birkhäuser Boston, Mass

    Google Scholar 

  9. J. Wiśniewski (1991) ArticleTitleOn contractions of extremal rays of Fano manifolds J. Reine Angew. Math 417 141–157

    Google Scholar 

  10. T. Ando (1985) ArticleTitleOn extremal rays of the higher dimensional varieties Invent. Math 81 347–357 Occurrence Handle10.1007/BF01389057

    Article  Google Scholar 

  11. M. Szurek J. Wiśniewski (1990) ArticleTitleFano bundles over \({\mathbb P} ^3\) and Q3 Pacific J Math 141 IssueID1 197–208

    Google Scholar 

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  1. Institut Fourier, UMR 5582, Université de Grenoble 1, BP 74, 38402, Saint Martin d’Hères, France

    Laurent Bonavero

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  1. Laurent Bonavero
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Correspondence to Laurent Bonavero.

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Bonavero, L. Pseudo-Index of Fano Manifolds and Smooth Blow-Ups. Geom Dedicata 114, 79–86 (2005). https://doi.org/10.1007/s10711-004-1816-1

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  • DOI: https://doi.org/10.1007/s10711-004-1816-1

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