Three-Dimensional Quartics and Counterexamples to the Lüroth Problem

Weighted Fano threefold hypersurfaces

“…We consider only the case N = 58 because the other cases are similar. Then X is a sufficiently general hypersurface of degree 24 in P (1,3,4,7,10). It is enough to show that the fiber C of the projection ψ over a point (p 1 : p 2 : p 3 ) ∈ P(1, 3, 4) is irreducible if p 1 = 0 and (p 1 : p 2 : p 3 ) belongs to the complement to a finite set.…”

“…Therefore, the properties described above can be naturally expected on anticanonically embedded quasismooth weighted Fano 3-fold hypersurfaces with terminal singularities. The first step in this direction is done in [10], where the birational superrigidity of smooth quartic 3-folds is proved.…”

Birational geometry of Fano hypersurfaces of index two

“…Note that the problem of description of the birational type of Fano varieties of index higher than one was discussed already in the classical paper [13]; Fano himself also worked on the problem (for the cubic threefold V 3 ⊂ P 4 ) [6].…”

A classification of terminal quartic 3-folds and applications to rationality questions

“…Intuitively, Fano varieties can be thought of as being close to P n : they are covered by rational curves and, in some sense, these curves should govern their birational geometry. However, the rationality question proved very difficult and it was not until the early seventies that it was settled for nonsingular Fano hypersurfaces in P 4 [IM71,CG72]. [IM71] developed the Noether-Fano method and proved that any smooth quartic hypersurface is birationally rigid -i.e.…”

“…Questions on rationality of Y , or at the other end of the spectrum, questions on rigidity, can be easier to answer on these models than on Y itself. A nonsingular quartic hypersurface Y = Y 4 ⊂ P 4 is nonrational [IM71]; more precisely, Y is the unique Mori fibre space in its birational equivalence class (up to birational automorphisms), i.e. Y 4 is birationally rigid.…”