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

Cited by in corpus (4)

• Rational curves on hypersurfaces
• Hilbert schemes of rational curves on Fano hypersurfaces
• An example on deformation of a pair
• The normal bundle of a rational curve on a generic quintic threefold