Rational curves on generic quintic threefolds

Abstract

Let $X_0$ be a generic quintic threefold in projective space $\mathbf P^4$ over complex numbers and $C_0$ be an irreducible rational curve on $X_0$. Let $$c_0: \mathbf P^1\to C_0\subset X_0$$ be its normalization. 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 $$H^1(N_{c_0/X_0})=0.$$