Uniform families of minimal rational curves on Fano manifolds
type: Item
instance of: scholarly article
title: Uniform families of minimal rational curves on Fano manifolds
zbMATH Open document ID: 1338.14043
author: Gianluca Occhetta
author: Kiwamu Watanabe
author: Luis E. Solá Conde
published in: Revista Matemática Complutense
publication date: 2016-05-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1503.05312
full work available at URL: http://oa.upm.es/43864/
review text: An interesting class of Fano manifolds \(X\) is that of rational homogeneous ones. Conjecturally (Campana-Peternell conjecture) they can be characterized as the Fano manifolds whose tangent is nef. In this context the rational curves on \(X\) are of interest and it is natural to try to reconstruct the homogeneous structure of \(X\) upon them (see the introduction of the paper under review and references therein). On the other hand the condition on the tangent bundle \(T_X\) to be nef imposes conditions on the possible splitting types with respect to rational curves on \(X\) which deserve to be studied. It would be interesting to know for instance if, choosen \(\mathcal M\) a locally unsplit dominating family of rational curves on \(X\), \(T_X\) is uniform (its restrictions to curves in \(\mathcal M\) are all isomorphic), being nef. The paper under review explores the consequences of \(T_X\) to be uniform on \(X\) Fano of Picard number one. In fact, see Theorem 1, if moreover one considers the minimal sections on \(\mathbb P(T_X)\) of curves in \(\mathcal M\) and imposes uniformity of the tangent to \(\mathbb P(T_X)\) with respect to them -- \(2\)-uniformity -- then \(X\) is rational homogeneous (of a particular type), up to a condition on the anticanonical degree of \(\mathcal M\) (which sounds natural in view of Hartshorne's conjecture on complete intersections). An application to Campana-Peternell conjecture (see Cor. 2) is also provided. To be precise: a Fano manifold of Picard number one whose tangent is nef and big, containing a family in the same conditions of Theorem 1, is rational homogeneous under the following condition: for \(\epsilon\) the birational crepant contraction of \(\mathbb P(T_X)\) associated to \(\mathcal O(1)\), its restriction to its exceptional locus is a smooth morphism.
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14M22
zbMATH DE Number: 6580075
zbMATH Keywords: Fano manifolds
zbMATH Keywords: homogeneity
zbMATH Keywords: VMRT
zbMATH Keywords: dual varieties
reviewed by: Roberto Muñoz
MaRDI profile type: Publication
OpenAlex ID: W2964048802
arXiv ID: 1503.05312
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DOI: 10.1007/S13163-015-0183-9
description: scientific article; zbMATH DE number 6580075