# Linear subspaces of hypersurfaces

## TL;DR

This paper proves conjectures about the dimensions of spaces of linear subspaces on smooth hypersurfaces in complex projective space, establishing new bounds for lines and k-planes, and exploring implications for unirationality and irreducibility of curve spaces.

## Contribution

It proves the de Jong-Debarre Conjecture for lines and extends results to k-planes, providing new bounds for the irreducibility and dimension of these subspace spaces.

## Key findings

- Dimension of lines in hypersurfaces is 2n-d-3 for n ≥ 2d-4.
- Space of k-planes on hypersurfaces is irreducible with expected dimension under certain bounds.
- Hypersurfaces with sufficiently large n are unirational, and spaces of degree e curves are irreducible under specified conditions.

## Abstract

Let $X$ be an arbitrary smooth hypersurface in $\mathbb{C} \mathbb{P}^n$ of degree $d$. We prove the de Jong-Debarre Conjecture for $n \geq 2d-4$: the space of lines in $X$ has dimension $2n-d-3$. We also prove an analogous result for $k$-planes: if $n \geq 2 \binom{d+k-1}{k} + k$, then the space of $k$-planes on $X$ will be irreducible of the expected dimension. As applications, we prove that an arbitrary smooth hypersurface satisfying $n \geq 2^{d!}$ is unirational, and we prove that the space of degree $e$ curves on $X$ will be irreducible of the expected dimension provided that $d \leq \frac{e+n}{e+1}$.