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Are minimal degree rational curves determined by their tangent vectors? (English) Zbl 1067.14023
Let $$X$$ be a projective variety, over an algebraically closed field, that is covered by rational curves (e.g. a Fano manifold over $${\mathbb Z}$$). In [J. Algebr. Geom. 11, 245–256 (2002; Zbl 1054.14035)] the first author considered the question of when there exists a unique minimal degree rational curve containing two given points. This paper was motivated by the following infinitesimal analogue: Are there natural conditions that guarantee that a minimal degree rational curve is uniquely determined by a tangent vector? Following Miyaoka’s approach, the authors show that in characteristic 0, if $$H \subset \text{RatCurves}^n(X)$$ is a proper, covering family of rational curves such that none of the associated curves has a cuspidal singularity, and if $$x \in X$$ is a general point, then all curves associated with the closed subfamily $$H_x := \{ \ell \in H \mid x \in \ell \} \subset H$$ are smooth at $$x$$, and no two of them share a common tangent direction at $$x$$. For positive characteristic an additional hypothesis is needed to obtain this conclusion. A central element of the proof is the study of families of dubbies, that is, reducible curves that consist of touching rational curves. Three applications are given. First, under certain hypotheses, $$H_x$$ is irreducible. For the second application, let $$X$$ be a complex variety satisfying the above hypotheses, and assume also that $$b_2(X) =1$$. Let $$\text{Aut}_0(X)$$ denote the maximal connected subgroup of the group of automorphisms of $$X$$, and let $$\text{Aut}_0(H)$$ be the analogous automorphism group for $$H$$. Then these groups coincide. The third application is that the main results apply also to contact manifolds.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14J45 Fano varieties 14H45 Special algebraic curves and curves of low genus
##### Keywords:
Fano manifold; rational curve of minimal degree; dubbies
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##### References:
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