Deformation classes of real four-dimensional cubic hypersurfaces
Authors:
S. Finashin and V. Kharlamov
Journal:
J. Algebraic Geom. 17 (2008), 677-707
DOI:
https://doi.org/10.1090/S1056-3911-08-00491-8
Published electronically:
March 3, 2008
MathSciNet review:
2424924
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We study real nonsingular cubic hypersurfaces $X\subset P^5$ up to deformation equivalence combined with projective equivalence and prove that they are classified by the conjugacy classes of involutions induced by the complex conjugation in $H_4(X)$. Moreover, we provide a graph $\Gamma _{K4}$ whose vertices represent the equivalence classes of such cubics and whose edges represent their adjacency. It turns out that the graph $\Gamma _{K4}$ essentially coincides with the graph $\Gamma _{K3}$ characterizing a certain adjacency of real nonpolarized K3-surfaces.
References
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References
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- A. Degtyarev, I. Itenberg and V. Kharlamov, Real Enriques surfaces, Lecture Notes Math., Springer 1746 (2000), 259 pages. MR 1795406 (2001k:14100)
- A. Degtyarev, I. Itenberg and V. Kharlamov, Finiteness and Quasi-Simplicity for Symmetric K3-Surfaces, Duke Math. J. 122 (2004), 1–49. MR 2046806 (2005d:14055)
- T. Fujita, Classification Theorems of Polarized Varieties, Lecture Notes Series, London Math. Society, vol. 155, 1990.
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- C. Voisin, Théorème de Torelli pour les cubiques de $P^5$, Invent. Math. 86 (1986), 577–601. MR 860684 (88g:14006)
- C. T. C. Wall, Quadratic forms in finite groups and related topics, Topology 2 (1964), 281–298. MR 0156890 (28:133)
Additional Information
S. Finashin
Affiliation:
Middle East Technical University, Department of Mathematics, Ankara 06531 Turkey
MR Author ID:
244559
Email:
serge@metu.edu.tr
V. Kharlamov
Affiliation:
Louis Pasteur et IRMA (CNRS)\endgraf7 rue René Descartes 67084 Strasbourg Cedex, France
MR Author ID:
202474
ORCID:
0000-0001-9341-1391
Email:
kharlam@math.u-strasbg.fr
Received by editor(s):
July 5, 2006
Received by editor(s) in revised form:
May 2, 2007
Published electronically:
March 3, 2008
Additional Notes:
The second author is supported by ANR-05-BLAN-0053-01.
