Projective manifolds containing special curves

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概要

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• Let Y be a smooth curve embedded in a complex projective manifold X of dimension n&\ge;2 with ample normal bundle NY|X. For every p&\ge;0 let αp denote the natural restriction maps Pic(X)→Pic(Y(p)), where Y(p) is the p-th infinitesimal neighbourhood of Y in X. First one proves that for every p&\ge;1 there is an isomorphism of abelian groups Coker(αp)$\cong$Coker(α0)$\oplus$Kp(Y,X), where Kp(Y,X) is a quotient of the C-vector space Lp(Y,X):=$\bigoplus$i=1p H1(Y, Si(NY|X)*) by a free subgroup of Lp(Y,X) of rank strictly less than the Picard number of X. Then one shows that L1(Y,X)=0 if and only if Y$\cong$P1 and NY|X$\cong\mathscr{O}_{\bm{P}^1}(1)^{\oplus n-1}$ (i.e. Y is a quasi-line in the terminology of [4]). The special curves in question are by definition those for which dimCL1(Y,X)=1. This equality is closely related with a beautiful classical result of B. Segre [25]. It turns out that Y is special if and only if either Y$\cong$P1 and NY|X$\cong\mathscr{O}_{\bm{P}^1}(2)\oplus\mathscr{O}_{\bm{P}^1}(1)^{\oplus n-2}$, or Y is elliptic and deg(NY|X)=1. After proving some general results on manifolds of dimension n≥2 carrying special rational curves (e.g. they form a subclass of the class of rationally connected manifolds which is stable under small projective deformations), a complete birational classification of pairs (X,Y) with X surface and Y special is given. Finally, one gives several examples of special rational curves in dimension n≥3.

• 2006-01-01

著者

• Beltrametti Mauro

  Universita Degli Studi Di Genova Dipartimento Di Matematica

• BADESCU Lucian

  Universita degli Studi di Genova Dipartimento di Matematica

関連論文

• Projective manifolds containing special curves

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