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Projective manifolds containing special curves | | Lucian Badescu
; Mauro C. Beltrametti
; | | Date: |
6 Feb 2007 | | Journal: | J. Math. Soc. Japan Vol. 58, No. 1, 2006, 211-230 | | Subject: | Algebraic Geometry | | Abstract: | Let $Y$ be a smooth curve embedded in a complex projective manifold $X$ of dimension $ngeq 2$ with ample normal bundle $N_{Y X}$. For every $pgeq 0$ let $alpha_p$ denote the natural restriction maps $Pic(X) oPic(Y(p))$, where $Y(p)$ is the $p$-th infinitesimal neighbourhood of $Y$ in $X$. First one proves that for every $pgeq 1$ there is an isomorphism of abelian groups $Coker(gra_p)congCoker(gra_0)oplus K_p(Y,X)$, where $K_p(Y,X)$ is a quotient of the $mathbb C$-vector space $L_p(Y,X):=igopluslimits_{i=1}^p H^1(Y, {f S}^i(N_{Y X})^*)$ 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 $Ycongmathbb P^1$ and $N_{Y X}congmathcal O_{mathbb P^1}(1)^{oplus n-1}$. 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 $Ycongmathbb P^1$ and $N_{Y X}congsO_{pn 1}(2)oplussO_{pn 1}(1)^{oplus n-2}$, or $Y$ is elliptic and $deg(N_{Y X})=1$. After proving some general results on manifolds of dimension $ngeq 2$ 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 $ngeq 3$. | | Source: | arXiv, math/0702148 | | Services: | Forum | Review | PDF | Favorites | |
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