Families of singular rational curves.

*(English)*Zbl 1054.14035From the introduction: This work is concerned with the study of algebraic families of rational curves. The main theorem yields a splitting-criterion for families of singular curves. In some cases, this effectively complements the bend-and-break argument which appears in the work of Mori. This result is applied to projective varieties \(X\) which are covered by a family of rational curves of minimal degrees. More precisely, for a fixed general point \(x\in X\), the author proves a sharp bound for the dimension of the subfamily of singular rational curves which pass through \(x\). Furthermore, he describes the singularities of these curves. As an application the author shows that the tangent map that sends a curve through \(x\) to its tangent direction in \(\mathbb{P}(T^*_X|_x)\) is a finite morphism, which may be useful in the further study of Fano manifolds with Picard number one.

A second application relates to Mori’s bend-and-break argument, which asserts that if \(x,y\in X\) are two general points, then there are at most finitely many curves in the family which contain both \(x\) and \(y\). In this work the author sheds some light on the question as to whether two sufficiently general points actually define a unique curve.

Finally, he gives a characterization of the projective space, which improves on some of the known generalizations of Kobayashi-Ochiai’s theorem in the characteristic 0 case.

A second application relates to Mori’s bend-and-break argument, which asserts that if \(x,y\in X\) are two general points, then there are at most finitely many curves in the family which contain both \(x\) and \(y\). In this work the author sheds some light on the question as to whether two sufficiently general points actually define a unique curve.

Finally, he gives a characterization of the projective space, which improves on some of the known generalizations of Kobayashi-Ochiai’s theorem in the characteristic 0 case.

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[5] | J. KollĂˇr. Rational Curves on Algebraic Varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. Springer, 1996. |

[6] | Y. Kachi and E. Sato. Polarized varieties whose points are joined by rational curves of small degrees. Illinois J. Math., 43(2):350-390, 1999. · Zbl 0939.14008 |

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