Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 58, Number 1 (2006), 211-230.
Projective manifolds containing special curves
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 .
J. Math. Soc. Japan, Volume 58, Number 1 (2006), 211-230.
First available in Project Euclid: 17 April 2006
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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
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. https://projecteuclid.org/euclid.jmsj/1145287099