×

zbMATH — the first resource for mathematics

Real congruence of complex matrix pencils and complex projections of real Veronese varieties. (English) Zbl 1049.14042
The paper treats the classification problem of homogeneous quadratic maps of a real projective space to a complex projective space via its relation to the congruence problem of pencils of symmetric matrices and the real classification of complex quadrics. The low-dimensional examples of quadratic curves in the Riemann sphere and of the models of the real projective plane in the complex four-space are considered in detail, resulting in a list of normal forms of real submanifolds of complex spaces with CR singularities.

MSC:
14P05 Real algebraic sets
14B05 Singularities in algebraic geometry
51N15 Projective analytic geometry
15A22 Matrix pencils
32V40 Real submanifolds in complex manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agafonov, S.; Ferapontov, E., Systems of conservation laws of temple class, equations of associativity and linear congruences in \( P\^{}\{4\}\), Manuscr. math., 106, 4, 461-488, (2001) · Zbl 1149.35385
[2] Apéry, F., Models of the real projective plane, (1987), Vieweg Braunschweig
[3] Boothby, W., An introduction to differentiable manifolds and Riemannian geometry, Pure and applied mathematics, vol. 120, (1986), Academic Press Boston · Zbl 0596.53001
[4] Coffman, A., CR singular immersions of complex projective spaces, Beiträge algebra geomet., 43, 2, 451-477, (2002) · Zbl 1029.32020
[5] A. Coffman, Formal stability of the CR cross-cap, Pacific J. Math., in press · Zbl 1123.32018
[6] A. Coffman, Analytic normal form for CR singular surfaces in \(C\^{}\{3\}\), preprint · Zbl 1074.32013
[7] A. Coffman, Notes on Abstract Linear Algebra, unpublished notes
[8] A. Coffman, M. Frantz, Möbius transformations and ellipses, preprint
[9] Coffman, A.; Schwartz, A.; Stanton, C., The algebra and geometry of Steiner and other quadratically parametrizable surfaces, Comput. aid. geomet. des., 13, 3, 257-286, (1996) · Zbl 0875.68860
[10] Corbas, B.; Williams, G.D., Congruence of two-dimensional subspaces in M2(k) (characteristic≠2), Pacific J. math., 188, 2, 225-235, (1999) · Zbl 0929.16029
[11] Cox, D.; Little, J.; O’Shea, D., Ideals, varieties, and algorithms, undergraduate texts in mathematics, (1992), Springer New York
[12] W.L.F. Degen, The types of triangular Bézier surfaces, in: G. Mullineux (Ed.), The Mathematics of Surfaces VI, Inst. Math. Appl. Conf. Ser., New Ser. 58, Oxford, 1996, pp. 153-170 · Zbl 0876.68112
[13] A. Degtyarev, Quadratic transformations \(RP\^{}\{2\}→RP\^{}\{2\}\), in: Topology of Real Algebraic Varieties and Related Topics, AMS Transl. Ser. 2, 173, Providence, 1996, pp. 61-71
[14] Forstnerič, F., Some totally real embeddings of three-manifolds, Manuscr. math., 55, 1, 1-7, (1986) · Zbl 0586.32027
[15] Forstnerič, F., Complex tangents of real surfaces in complex surfaces, Duke math. J., 67, 2, 353-376, (1992) · Zbl 0761.53032
[16] Harris, J., Algebraic geometry: a first course, GTM 133, (1992), Springer New York
[17] Kasner, E., The invariant theory of the inversion group: geometry upon a quadric surface, Trans. am. math. soc., 1, 4, 430-498, (1900) · JFM 31.0652.02
[18] Levy, H., Projective and related geometries, (1967), Macmillan New York
[19] Morley, F.; Morley, F.V., Inversive geometry, (1954), Chelsea New York · Zbl 0009.02908
[20] Morley, F.; Patterson, B., On algebraic inversive invariants, Am. J. math., 52, 2, 413-424, (1930) · JFM 56.0123.03
[21] Moser, J., Analytic surfaces in \(C\^{}\{2\}\) and their local hull of holomorphy, Ann. acad. scient. fenn., ser. A. I., 10, 397-410, (1985) · Zbl 0585.32007
[22] Patterson, B., The inversive plane, Am. math. monthly, 48, 9, 589-599, (1941) · Zbl 0060.33108
[23] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, vol. II, fifth ed. (with contributions by G. Webb), Chelsea, New York, 1965
[24] Searle, S., Matrix algebra useful for statistics, (1982), Wiley New York · Zbl 0555.62002
[25] D. Sommerville, Analytical Geometry of Three Dimensions, Cambridge, 1934 · Zbl 0008.40203
[26] Waterhouse, W., Real classification of complex quadrics, Linear algebra appl., 48, 45-52, (1982) · Zbl 0503.51018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.