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# Asian Journal of Mathematics

## Volume 18 (2014)

### Number 1

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

Pages: 101 – 116

DOI: http://dx.doi.org/10.4310/AJM.2014.v18.n1.a5

#### Author

#### 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.

#### Keywords

rational curve, hypersurface, twisted normal bundle

#### 2010 Mathematics Subject Classification

14J70