Journal of the Mathematical Society of Japan

Projective manifolds containing special curves


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Let Y be a smooth curve embedded in a complex projective manifold X of dimension n2 with ample normal bundle NY|X . For every p0 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 p1 there is an isomorphism of abelian groups Coker ( α p ) Coker ( α 0 ) K p ( Y , X ) , where Kp(Y,X) is a quotient of the C-vector space L p ( Y , X ) : = i = 1 p H 1 ( Y , S i ( N Y | X ) * ) by a free subgroup of L p ( Y , X ) of rank strictly less than the Picard number of X. Then one shows that L 1 ( Y , X ) = 0 if and only if YP1 and N Y | X 𝒪 P 1 ( 1 ) 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 dim C L1(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 YP1 and NY|X𝒪P 1(2) 𝒪 P 1 ( 1 ) 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.

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J. Math. Soc. Japan, Volume 58, Number 1 (2006), 211-230.

First available in Project Euclid: 17 April 2006

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Zentralblatt MATH identifier

Primary: 14C22: Picard groups 14H45: Special curves and curves of low genus 14D15: Formal methods; deformations [See also 13D10, 14B07, 32Gxx]
Secondary: 14E99: None of the above, but in this section

curves with ample normal bundle in a manifold infinitesimal neighbourhood Picard group rationallly connected varieties


BĂDESCU, Lucian; BELTRAMETTI, Mauro C. Projective manifolds containing special curves. J. Math. Soc. Japan 58 (2006), no. 1, 211--230. doi:10.2969/jmsj/1145287099.

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