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 $e \leq n$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2017
- DOI:
- 10.48550/arXiv.1705.08441
- arXiv:
- arXiv:1705.08441
- Bibcode:
- 2017arXiv170508441C
- Keywords:
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- Mathematics - Algebraic Geometry
- E-Print:
- Comments welcome!