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Uniform families of minimal rational curves on Fano manifolds

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Abstract

It is a well known fact that families of minimal rational curves on rational homogeneous manifolds of Picard number one are uniform, in the sense that the tangent bundle to the manifold has the same splitting type on each curve of the family. In this note we prove that certain—stronger—uniformity conditions on a family of minimal rational curves on a Fano manifold of Picard number one allow to prove that the manifold is homogeneous.

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Acknowledgments

The results in this paper were obtained mostly while the second author was a Visiting Researcher at the Korea Institute for Advanced Study (KIAS) and the Department of Mathematics of the University of Warsaw. He would like to thank both institutions for their support and hospitality. The authors would like to thank J. Wiśniewski for his interesting comments and discussions on this topic, and an anonymous referee, whose remarks helped to improve substantially the final form of this paper.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38123, Povo di Trento, TN, Italy

    Gianluca Occhetta & Luis E. Solá Conde

  2. Course of Mathematics, Programs in Mathematics, Electronics and Informatics, Graduate School of Science and Engineering, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama, 338-8570, Japan

    Kiwamu Watanabe

Authors
  1. Gianluca Occhetta
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  2. Luis E. Solá Conde
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  3. Kiwamu Watanabe
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Correspondence to Gianluca Occhetta.

Additional information

G. Occhetta partially supported by PRIN project “Geometria delle varietà algebriche” and the Department of Mathematics of the University of Trento. L. E. S. Conde supported by the Korean National Researcher Program 2010-0020413 of NRF, and by the Polish National Science Center project 2013/08/A/ST1/00804. Kiwamu Watanabe: partially supported by JSPS KAKENHI Grant Number 26800002.

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Occhetta, G., Solá Conde, L.E. & Watanabe, K. Uniform families of minimal rational curves on Fano manifolds. Rev Mat Complut 29, 423–437 (2016). https://doi.org/10.1007/s13163-015-0183-9

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