Projective manifolds containing special curves

Abstract

Let $Y$ be a smooth curve embedded in a complex projective manifold $X$ of dimension $n\ge 2$ with ample normal bundle $N_{Y|X}$. For every $p\ge 0$ let ${\alpha }_p$ denote the natural restriction maps $\Pic(X)\to\Pic(Y(p))$, where $Y(p)$ is the $p$-th infinitesimal neighbourhood of $Y$ in $X$. First one proves that for every $p\ge 1$ there is an isomorphism of abelian groups $\Coker(\gra_p)\cong\Coker(\gra_0)\oplus K_p(Y,X)$, where $K_p(Y,X)$ is a quotient of the $\mathbb{C}$-vector space $L_p(Y,X):=\underset{i=1}{\overset{p}{⨁}}{H}^1(Y,{\mathbf{S}}^i(N_{Y|X}{)}^{\ast })$ 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 $Y\cong {\mathbb{P}}^1$ and $N_{Y|X}\cong {\mathcal{O}}_{{\mathbb{P}}^1}(1{)}^{\oplus n-1}$. The special curves in question are by definition those for which ${\dim }_{\mathbb{C}}{L}_1(Y,X)=1$. This equality is closely related with a beautiful classical result of B. Segre. It turns out that $Y$ is special if and only if either $Y\cong {\mathbb{P}}^1$ and $N_{Y|X}\cong\sO_{\pn 1}(2)\oplus\sO_{\pn 1}(1)^{\oplus n-2}$, or $Y$ is elliptic and $°(N_{Y|X})=1$. After proving some general results on manifolds of dimension $n\ge 2$ 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 $n\ge 3$.