# Normal bundles of lines on hypersurfaces

## TL;DR

This paper studies the stratification of the Fano scheme of lines on smooth hypersurfaces based on the splitting types of their normal bundles, providing dimension and class computations for general hypersurfaces.

## Contribution

It introduces a stratification of lines on hypersurfaces by normal bundle splitting types and computes their classes and dimensions for general cases.

## Key findings

- For general hypersurfaces, the strata have the expected dimension.
- The class of the closure of each stratum is computed in the Chow ring.
- Upper bounds on the dimension of certain strata are established for all smooth hypersurfaces.

## Abstract

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 $X$.