Geodesic
Three-dimensional quartics and counterexamples to the L\"uroth problem
Sbornik. Mathematics, Tome 15 (1971) no. 1, pp. 141-166

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In this paper we prove that any birational mapping between smooth hypersurfaces of degree four is an isomorphism. Since B. Segre constructed examples of smooth unirational quartics, this leads to a negative resolution of the three-dimensioal Lüroth problem. Bibliography: 13 titles. 
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V. A. Iskovskikh; Yu. I. Manin. Three-dimensional quartics and counterexamples to the L\"uroth problem. Sbornik. Mathematics, Tome 15 (1971) no. 1, pp. 141-166. http://geodesic.mathdoc.fr/item/SM_1971_15_1_a7/