Abstract
Let X, Y be Fano threefolds of Picard number one and such that the ample generators of Picard groups are very ample. Let X be of index one and Y be of index two. It is shown that the only morphisms from X to Y are double coverings. In fact nearly the whole paper is the analysis of the case where Y is the linear section of the Grassmannian G(1,4) , since the other cases were more or less solved in another article. This remaining case is treated with the help of Debarre's connectedness theorem for inverse images of Schubert cycles.
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