Uniform families of minimal rational curves on Fano manifolds
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.
scientific article; zbMATH DE number 6580075
scientific article