Let $X\subset \mathbb P^{n+1}_{\mathbb C}$ be smooth hypersurface of degree $d$. Set $L(a_1,a_2):=\overline{\{([\ell],(x_1,x_2))\in Gr(2,n+2)\times X^2, x_i\in \ell\ and\ \{x_1\}^{a_1}\cup\{x_2\}^{a_2}\subset \ell\cap X\}}$
where the last condition means that $\ell\cap X$ contains $\{x_i\}$ with multiplicity $a_i>0$ (with $a_1+a_2 \leq d$).
It seems that the dimension ($2n+2-(a_1+a_2)$ for a general $X$) of $L(a_1,a_2)$ depends only on the sum $a_1+a_2$ when one tries to look at it as a zero locus of a vector bundle in $\{([\ell],(x_1,x_2))\in Gr(2,n+2)\times (\mathbb P^{n+1})^2, x_i\in \ell\}$. I do not understand the intuition behind the result that lines whose intersection with $X$ "are of different weights" are "equally abundant".
So is my computation wrong?

  • 1
    $\begingroup$ Your computation is correct. One related fact has to do with the infinitesimal deformation theory of $L(a_1,a_2)$. For a given point $([\ell],(x_1,x_2))$ in $L(a_1,a_2)$, the vector space of first order deformation is $H^0(\ell,\mathcal{E})$ where $\mathcal{E}$ is the kernel of the sheaf homomorphism $$N_{\ell/\mathbb{P}^{n+1}}\to T_{\mathbb{P}^{n+1}}/(T_\ell + T_X)|_{a_1\underline{x}_1+a_2\underline{x}_2}.$$ Since the invertible sheaf $\mathcal{O}_\ell(-a_1\underline{x}_1-a_2\underline{x}_2)$ depends only on $a_1+a_2$, that helps explain this fact. $\endgroup$ – Jason Starr Jul 20 '17 at 11:35

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