Normal bundles of rational curves on complete intersections

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 $e\le n$ with balanced normal bundle. If all $d_i$ are $2$ and $n\ge 2k+1$, we show that $X$ contains rational curves of degree $e\le n-1$ 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.

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