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 $\text{Pic}\left(X\right)\to \text{Pic}\left(Y\right(p\left)\right)$, where $Y\left(p\right)$ 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 $\text{Coker}\left({\alpha }_p\right)\cong \text{Coker}\left({\alpha }_0\right)\oplus K_p(Y,X)$, where $K_p(Y,X)$ is a quotient of the $C$-vector space $L_p(Y,X):=\underset{i=1}{\overset{p}{⨁}}{H}^1(Y,{\mathbf{S}}^i(N_{Y|X}{)}^{*})$ 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 $Y\cong {\mathbf{P}}^{\mathbf{1}}$ and $N_{Y|X}\cong {𝒪}_{{\mathbf{P}}^{\mathbf{1}}}(1{)}^{\oplus n-1}$ (i.e. $Y$ is a quasi-line in the terminology of [4]). The special curves in question are by definition those for which ${\dim }_{\mathbf{C}}{L}_1(Y,X)=1$ . 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 $Y\cong {\mathbf{P}}^{\mathbf{1}}$ and $N_{Y|X}\cong {𝒪}_{{\mathbf{P}}^{\mathbf{1}}}\left(2\right)\oplus {𝒪}_{{\mathbf{P}}^{\mathbf{1}}}(1{)}^{\oplus n-2}$ , or $Y$ is elliptic and $\mathrm{deg}\left(N_{Y|X}\right)=1$ . 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 $n\ge 3$.