First order deformations of pairs of rational curves and quintic threefolds

Details First order deformations of pairs of rational curves and quintic threefolds First order deformations of pairs of rational curves and quintic threefolds

2012-02-13

arxiv.org/abs/1202.2831v1

Mathematics

Algebraic Geometry

Scientific paper

This is one of a series of papers ([3], [4], [5]) that study the pairs of curves and hypersurfaces under the same motif: a fundamental property in the first order deformations of the pairs (the condition (1.2) below) has a deep effect on the structure of the normal sheaves of the curves in the hypersurfaces. Each paper solves an independent problem with the same condition (1.2), which addresses only one aspect of the normal sheaves. In this paper, we let $X_0$ be a specific type of smooth quintic threefolds (a generic quintic is this type) in projective space $\mathbf P^4$ over complex numbers, which is called a "totally non-degenerated" quintic threefold (see the definition (1.2)), and $C_0$ an irreducible rational curve on $f_0$. We prove that if the first order deformation of the pair $ C_0, X_0$ exists along each deformation of the hypersurface $X_0$, i.e. the condition (1.2) below holds, then $$H^1(N_{c_0/X_0})=0,$$ where $c_0: \mathbf P^1\to \mathbf C_0$ is a normalization of $C_0$.

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