Let $X \subset \mathbb{ P}^n$ be a smooth hypersurface. Given a sequence of integers $\vec{ a} = (a_1, \ldots, a_{ n-2})$ with $a_1 \leq \cdots \leq a_{ n-2}$, 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 $X$.
Categories: math.AG