Rocky Mountain Journal of Mathematics

First order deformations of pairs and non-existence of rational curves

Bin Wang

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Let $X_0$ be a smooth hypersurface (assumed not to be generic) in projective space $\mathbf {P}^n$, $n\geq 4$, over complex numbers, and $C_0$ a smooth rational curve on $X_0$. We are interested in deformations of the pair $C_0$ and $X_0$. In this paper, we prove that, if the first order deformations of the pair exist along each deformation of the hypersurface $X_0$, then $\deg (C_0)$ cannot be in the range \[ \bigg ( m\frac {2\deg (X_0)+1}{\deg (X_0)+1}, \frac {2+m(n-2)}{2n-\deg (X_0)-1}\bigg ), \] where $m$ is any non negative integer less than \[ \dim (H^0(\mathcal {O}_{\mathbf {P}^n}(1))|_{C_0} )-1. \]

Article information

Rocky Mountain J. Math., Volume 46, Number 2 (2016), 663-678.

First available in Project Euclid: 26 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J70: Hypersurfaces 14N10: Enumerative problems (combinatorial problems) 14N25: Varieties of low degree

Hypersurface rational curve normal bundle


Wang, Bin. First order deformations of pairs and non-existence of rational curves. Rocky Mountain J. Math. 46 (2016), no. 2, 663--678. doi:10.1216/RMJ-2016-46-2-663.

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