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JANUARY, 2006 Projective manifolds containing special curves
J. Math. Soc. Japan 58(1): 211-230 (JANUARY, 2006). DOI: 10.2969/jmsj/1145287099


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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Mauro C. BELTRAMETTI. Lucian BĂDESCU. "Projective manifolds containing special curves." J. Math. Soc. Japan 58 (1) 211 - 230, JANUARY, 2006.


Published: JANUARY, 2006
First available in Project Euclid: 17 April 2006

zbMATH: 1097.14008
MathSciNet: MR2204571
Digital Object Identifier: 10.2969/jmsj/1145287099

Primary: 14C22 , 14D15 , 14H45
Secondary: 14E99

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

Rights: Copyright © 2006 Mathematical Society of Japan


Vol.58 • No. 1 • JANUARY, 2006
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