## Journal of the Mathematical Society of Japan

### Projective manifolds containing special curves

#### Abstract

Let $Y$ be a smooth curve embedded in a complex projective manifold $X$ of dimension $n\geq 2$ with ample normal bundle $N_{Y|X}$ . For every $p\geq 0$ let $\alpha_p$ denote the natural restriction maps ${\rm Pic}(X)\to{\rm Pic}(Y(p))$, where $Y(p)$ is the $p$-th infinitesimal neighbourhood of $Y$ in $X$. First one proves that for every $p\geq 1$ there is an isomorphism of abelian groups ${\rm Coker}(\alpha_p)\cong{\rm Coker}(\alpha_0)\oplus K_p(Y,X)$, where $K_p(Y,X)$ is a quotient of the $\bm{C}$-vector space $L_p(Y,X):=\bigoplus_{i=1}^p H^1(Y, {\bf S}^i(N_{Y|X})^*)$ 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 $Y\cong\bm{P}^1$ and $N_{Y|X}\cong\mathscr{O}_{\bm{P}^1}(1)^{\oplus n-1}$ (i.e. $Y$ is a quasi-line in the terminology of [4]). The special curves in question are by definition those for which $\dim_{\bm{C}}L_1(Y,X)=1$ . 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 $Y\cong\bm{P}^1$ and $N_{Y|X}\cong\mathscr{O}_{\bm{P}^1}(2)\oplus\mathscr{O}_{\bm{P}^1}(1)^{\oplus n-2}$ , or $Y$ is elliptic and $\deg(N_{Y|X})=1$ . After proving some general results on manifolds of dimension $n\geq 2$ 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 $n\geq 3$.

#### Article information

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

Dates
First available in Project Euclid: 17 April 2006

https://projecteuclid.org/euclid.jmsj/1145287099

Digital Object Identifier
doi:10.2969/jmsj/1145287099

Mathematical Reviews number (MathSciNet)
MR2204571

Zentralblatt MATH identifier
1097.14008

#### Citation

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

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