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 $n\geq 3$ .