Hilbert scheme of rational curves on a generic quintic threefold

2018/12/06 by Wang, B.
#14J33 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.02526

Abstract

Let X₀ be a generic quintic threefold in projective space \mathbf P⁴ over the complex numbers. For a fixed natural number d, let R_d(X₀) be the open sub-scheme of the Hilbert scheme, parameterizing irreducible rational curves of degree d on X₀. In this paper, we show that (1) R_d(X₀) is smooth and of expected dimension, \par (2) Combining the Calabi-Yau condition on X₀, we further show that it consists of immersed rational curves. (3) Parts (1) and (2) imply a statement of Clemens' conjecture: if C₀∈ R_d(X₀) and c₀:\mathbf P¹→ C₀ is the normalization, the \par\hspace1cc normal sheaf is isomorphic to the vector bundle N_c0/X0≃ \mathcal O_{\mathbf P¹}(-1)⊕ \mathcal O_{\mathbf P¹}(-1).

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