Projective manifolds containing special curves
| dc.creator | Badescu, Lucian | |
| dc.creator | Beltrametti, Mauro C. | |
| dc.date | 2007-02-06 | |
| dc.date.accessioned | 2026-07-07T07:45:05Z | |
| dc.date.available | 2026-07-07T07:45:05Z | |
| dc.description | 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 $α_p$ 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 $p\geq 1$ 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 $L_p(Y,X):=\bigoplus\limits_{i=1}^p H^1(Y, {\bf S}^i(N_{Y|X})^*)$ 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 $Y\cong\mathbb P^1$ 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 $n\geq 2$ 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 $n\geq 3$. | |
| dc.identifier | https://arxiv.org/abs/math/0702148 | |
| dc.identifier | http://arxiv.org/abs/math/0702148 | |
| dc.identifier | J. Math. Soc. Japan Vol. 58, No. 1, 2006, 211-230 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/123425 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14C22, 14H45, 14D15 | |
| dc.title | Projective manifolds containing special curves | |
| dc.type | text |