Let be a smooth curve embedded in a complex projective manifold of dimension with ample normal bundle . For every let denote the natural restriction maps , where is the -th infinitesimal neighbourhood of in . First one proves that for every there is an isomorphism of abelian groups , where is a quotient of the -vector space by a free subgroup of of rank strictly less than the Picard number of . Then one shows that if and only if and (i.e. is a quasi-line in the terminology of ). The special curves in question are by definition those for which . This equality is closely related with a beautiful classical result of B. Segre . It turns out that is special if and only if either and , or is elliptic and . After proving some general results on manifolds of dimension 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 with surface and special is given. Finally, one gives several examples of special rational curves in dimension .
Mauro C. BELTRAMETTI. Lucian BĂDESCU. "Projective manifolds containing special curves." J. Math. Soc. Japan 58 (1) 211 - 230, JANUARY, 2006. https://doi.org/10.2969/jmsj/1145287099