We study the family of smooth rational curves of degree e on a hypersurface of degree d in ℙn. When e = 1, the smoothness, the dimension and the connectedness of the family have been investigated by W. Barth and A. Van de Ven over the complex number field, and by J. Kollár over an algebraically closed field of arbitrary characteristic. On the other hand, J. Harris, M. Roth, and J. Starr have recently studied irreducibility, smoothness and the dimension of the family over the complex number field. In this paper, we investigate the family in detail, mainly in the case of 1 ≦ e ≦ 3 and d ≧ 1, and in the case of e ≧ 4 and d ≧ 2e − 3, without the assumption on the characteristic of the ground field.
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