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

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 $(c_0)\ast: T{t}\mathbf P^1 \to T_{c_0(t)} X_0$isinjectiveateach$t\in \mathbf P^1$, (2) the normal bundle of $c_0$ satisfies $$H^1(N_{c_0\mathrm{/}{X}_0})=0.$$