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Families of singular rational curves. (English) Zbl 1054.14035
From 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.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14H20 Singularities of curves, local rings
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##### References:
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