# Hilbert scheme of rational curves on a generic quintic threefold

## TL;DR

This paper proves that the Hilbert scheme of rational curves on a generic quintic threefold is smooth, of expected dimension, consists of immersed curves, and confirms a case of Clemens' conjecture regarding the normal sheaf structure.

## Contribution

It establishes smoothness, expected dimension, and immersion properties of rational curves on a generic quintic threefold, confirming a specific case of Clemens' conjecture.

## Key findings

- Hilbert scheme of rational curves is smooth and of expected dimension
- Rational curves are immersed due to Calabi-Yau condition
- Normal sheaf is isomorphic to O(-1)⊕O(-1) for normalized curves

## Abstract

Let $X_0$ be a generic quintic threefold in projective space $\mathbf P^4$ over the complex numbers. For a fixed natural number $d$, let $R_d(X_0)$ be the open sub-scheme of the Hilbert scheme, parameterizing irreducible rational curves of degree $d$ on $X_0$. In this paper, we show that (1) $R_d(X_0)$ is smooth and of expected dimension, \par (2) Combining the Calabi-Yau condition on $X_0$, we further show that it consists of immersed rational curves. (3) Parts (1) and (2) imply a statement of Clemens' conjecture: if $C_0\in R_d(X_0)$ and $c_0:\mathbf P^1\to C_0$ is the normalization, the \par\hspace{1cc} normal sheaf is isomorphic to the vector bundle $$N_{c_0/X_0}\simeq \mathcal O_{\mathbf P^1}(-1)\oplus \mathcal O_{\mathbf P^1}(-1).$$