Obstructions to the deformations of curves to other hypersurfaces

Details Obstructions to the deformations of curves to other hypersurfaces Obstructions to the deformations of curves to other hypersurfaces

2011-10-02

arxiv.org/abs/1110.0184v3

Mathematics

Algebraic Geometry

Scientific paper

We are interested in obstructions to the FIRST order deformation of a pair of a smooth hypersurface $f_0$ and a smooth curve $C_0$ contained in $f_0$. In the first half of the paper, we give necessary conditions for the pair to deform in the first order. In particular, for a rational curve $C_0$, this necessary condition is $$H^1(N_{C_0}f_0(1))=0.$$ In the second half, we apply the necessary conditions from the first half of the paper to study the geometry of smooth curves in hypersurfaces. The main application is for the case where $C_0$ is a rational curve.

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