First order deformations of pairs of a rational curve and a hypersurface

Abstract

Let $X_0$ be a smooth hypersurface (not assumed generic) in projective space $\mathrm{P}^n$, $n \geq 3$ over the complex numbers, and $C_0$ a smooth rational curve on $X_0$. We are interested in the deformations of the pair $C_0 , X_0$. In this paper, we prove that if the first order deformations of the pair exist along certain first order deformations of the hypersurface $X_0$, then the twisted normal bundle $N_{C_0/ X_0}(1) = N_{C_0 / X_0} \otimes \mathcal{O}_{\mathcal{P}^n} (1) \vert {}_{C_0}$ is generated by global sections.