# Rational curves on complete intersection Calabi-Yau 3-folds

## TL;DR

This paper proves that generic complete intersection Calabi-Yau 3-folds contain rational curves of all degrees, which are immersions with specific normal bundle properties, advancing understanding of their geometric structure.

## Contribution

It establishes the existence of rational curves of all degrees on generic complete intersection Calabi-Yau 3-folds and characterizes their normal bundles.

## Key findings

- Existence of rational curves of every degree on generic Calabi-Yau 3-folds.
- All such rational curves are immersions.
- Normal bundle of these curves is isomorphic to or P^1}(-1)\u00d7or P^1}(-1).

## Abstract

We prove the following results. If $X_3$ is a generic complete intersection Calabi-Yau 3-fold, (1) then for each natural number $d$ there exists a rational map \par\hspace{1 cc} $c\in Hom_{bir}(\mathbf P^1, X_3)$ of $deg(c(\mathbf P^1))=d$, (2) further more all such $c$ are immersions satisfying \begin{equation} N_{c(\mathbf P^1)/ X_3}\simeq \mathcal O_{\mathbf P^1}(-1)\oplus \mathcal O_{\mathbf P^1}(-1).