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Positive sheaves of differentials coming from coarse moduli spaces

Kelly Jabbusch;  Stefan Kebekus — 2011

Annales de l’institut Fourier

Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base Y ∘ , and suppose the family is non-isotrivial. If Y is a smooth compactification of Y ∘ , such that D : = Y ∖ Y ∘ is a simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along D . Viehweg and Zuo have shown that for some m > 0 , the m th symmetric power of this sheaf admits many sections. More precisely, the m th symmetric power contains an invertible...

Are rational curves determined by tangent vectors?

Stefan Kebekus;  Sándor J. Kovács — 2004

Annales de l’institut Fourier

Let X 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 X 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.

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