Abstract
We study here the projective varieties with the property that there exists a projective isomorphism between two of their generic hyperplane sections. The case of surfaces had already been studied by Fubini and Fano in the 1920s. The latter gave the list of all (possibly signular) surfaces with projectively isomorphic hyperplane sections. The proof, however, was essentially wrong. By means of a different approach, we are able to supply a proof of Fano's claims. Moreover, we show some general properties of varieties with projectively isomorphic hyperplane sections: they have uniruled hyperplane sections and are related to varieties with ‘small’ dual varieties. In particular we are able to conclude that threefolds with projectively isomorphic hyperplane sections either have rational sections or are P2-bundles over a curve.
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