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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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