Normal bundles of lines on hypersurfaces

Abstract

Let $X\subset {\mathbb{P}}^n$ be a smooth hypersurface. Given a sequence of integers $\vec{a}=(a_1,\dots ,a_{n-2})$ with $a_1\le \cdots \le a_{n-2}$, let $F_{\vec{a}}(X)$ be the parameter space of lines $L$ on $X$ such that $N_{L/X}\cong \mathcal{O}(a_1)\oplus \cdots \oplus \mathcal{O}(a_{n-2})$. The loci $F_{\vec{a}}(X)$ form a stratification of the Fano scheme of lines on $X$. We show that for general hypersurfaces, the $F_{\vec{a}}(X)$ have the expected dimension and, in this case, compute the class of $\overline{F_{\vec{a}}(X)}$ in the Chow ring of the Grassmannian of lines in ${\mathbb{P}}^n$. For certain splitting types $\vec{a}$, we also provide non-trivial upper bounds on the dimension of $F_{\vec{a}}(X)$ that hold for all smooth $X$.