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

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 $$H^1(N_{c_0/X_0})=0.$$

arXiv commentarXiv admin note: substantial text overlap with arXiv:1202.2831

References in corpus (1)

• Rational curves on generic quintic threefolds