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Rigid isotopy classification of real three-dimensional cubics. (English. Russian original) Zbl 1222.14125
Izv. Math. 70, No. 4, 731-768 (2006); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 4, 91-134 (2006).
The paper contains a rigid isotopy classification of nonsingular real cubic hypersurfaces in the four-dimensional projective space \({\mathbb P}^4\). The author proves that the space of nonsingular real cubic hypersurfaces in \({\mathbb P}^4\) has \(9\) connected components. These components are distinguished by the homology of the real point set of a hypersurface and the information on triviality or nontriviality of the \({\mathbb Z}/2\)-homology class realized by the real point set in the complex point set. The proof follows Klein’s approach and uses a relation between the cubic hypersurfaces in \({\mathbb P}^n\) and complete intersections of bi-degree \((2,3)\) in \({\mathbb P}^{n-1}\). The rigid isotopy classification of nonsingular real cubic hypersurfaces in \({\mathbb P}^4\) is closely related to a rigid isotopy classification of real canonical curves of genus \(4\) in \({\mathbb P}^3\). The latter classification is also obtained in the paper and contains \(8\) classes. A similar approach was recently used by S. Finashin and V. Kharlamov [J. Algebr. Geom. 17, No. 4, 677–707 (2008; Zbl 1225.14047; arXiv:0607137)] in order to obtain a classification of nonsingular real cubic hypersurfaces in \({\mathbb P}^5\) up to deformation equivalence combined with projective equivalence.

MSC:
14P25 Topology of real algebraic varieties
14N25 Varieties of low degree
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