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Let be a projective variety which is covered by rational curves, for instance a Fano
manifold over the complex numbers. In this paper, we give sufficient conditions which
guarantee that every tangent vector at a general point of is contained in at most one
rational curve of minimal degree. As an immediate application, we obtain irreducibility
criteria for the space of minimal rational curves.
We show that if the degree of a nonsingular projective variety is high enough, maximization of any of the most important numerical invariants, such as class, Betti number, and any of the Chern or middle Hodge numbers, leads to the same class of extremal varieties. Moreover, asymptotically (say, for varieties whose total Betti number is big enough) the ratio of any two of these invariants tends to a well-defined constant.
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