# Normal bundles of rational curves on complete intersections

## TL;DR

This paper investigates the existence of rational curves with balanced normal bundles on general Fano complete intersections, providing new results that strengthen the understanding of their rational connectivity.

## Contribution

It establishes the presence of balanced normal bundle rational curves of certain degrees on general Fano complete intersections, improving previous results on their rational connectivity.

## Key findings

- Existence of rational curves with balanced normal bundle for degree e ≤ n or n-1.
- Proof of stronger separable rational connectivity of X.
- Construction of very free rational curves of optimal degrees.

## Abstract

Let $X \subset \mathbb{P}^n$ be a general Fano complete intersection of type $(d_1,\dots, d_k)$. If at least one $d_i$ is greater than $2$, we show that $X$ contains rational curves of degree $e \leq n$ with balanced normal bundle. If all $d_i$ are $2$ and $n\geq 2k+1$, we show that $X$ contains rational curves of degree $e \leq n-1$ with balanced normal bundle. As an application, we prove a stronger version of the theorem of Z. Tian \cite{Tian}, Q. Chen and Y. Zhu \cite{ChenZhu} that $X$ is separably rationally connected by exhibiting very free rational curves in $X$ of optimal degrees.