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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
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