Rocky Mountain Journal of Mathematics

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

Bin Wang

Abstract

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

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

Dates
First available in Project Euclid: 26 July 2016

https://projecteuclid.org/euclid.rmjm/1469537480

Digital Object Identifier
doi:10.1216/RMJ-2016-46-2-663

Mathematical Reviews number (MathSciNet)
MR3529086

Zentralblatt MATH identifier
1356.14032

Citation

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. https://projecteuclid.org/euclid.rmjm/1469537480

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