The normal bundle of a rational curve on a generic quintic threefold

TL;DR

This paper provides a new proof that rational curves on a generic quintic threefold are immersions with vanishing first cohomology of their normal bundle, confirming their expected geometric properties.

Contribution

It offers an alternative proof of the normal bundle properties of rational curves on generic quintic threefolds, emphasizing their immersion and cohomological vanishing.

Findings

Rational curves are immersions on generic quintic threefolds.

The normal bundle of such curves has vanishing first cohomology.

The proof aligns with and reinforces previous results.

Abstract

This is another proof of the same result in [9]. Let $X_0$ be a generic quintic hypersurface in $\mathbf P^4$ over $\mathbb C$ and $c_0$ a regular map $\mathbf P^1\to X_0$ that is generically one-to-one to its image. In this paper, we show (1) $c_0$ must be an immersion, i.e. the differential $(c_0)_\ast: T_{t}\mathbf P^1 \to T_{c_0(t)} X_0$ is injective at each $t\in \mathbf P^1$, (2) the normal bundle of $c_0$ satisfies $$H^1(N_{c_0/X_0})=0.$$