Displaying similar documents to “Enumerative geometry of divisorial families of rational curves”

Are rational curves determined by tangent vectors?

Stefan Kebekus, Sándor J. Kovács (2004)

Annales de l’institut Fourier

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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.

Generalized Mukai conjecture for special Fano varieties

Marco Andreatta, Elena Chierici, Gianluca Occhetta (2004)

Open Mathematics

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Let X be a Fano variety of dimension n, pseudoindex i X and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(i X−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i X≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρX> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension...

Rational equivalence on some families of plane curves

Josep M. Miret, Sebastián Xambó Descamps (1994)

Annales de l'institut Fourier

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If V d , δ denotes the variety of irreducible plane curves of degree d with exactly δ nodes as singularities, Diaz and Harris (1986) have conjectured that Pic ( V d , δ ) is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that Pic ( V d , 1 ) is a finite group, so that the conjecture holds for δ = 1 . Actually the order of Pic ( V d , 1 ) is 6 ( d - 2 ) d 2 - 3 d + 1 ) , the group being cyclic if d is odd and the product of ℤ 2 and a cyclic group of order 3 ( d - 2 ) ( d 2 - 3 d + 1 ) if d is even.