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Nilpotent algebras and affinely homogeneous surfaces. (English) Zbl 1254.14053
To a nilpotent commutative algebra \(N\) of finite dimension over a field of zero characteristic one associates a smooth algebraic subvariety \(S \subset N\), whose degree is the nil-index and whose codimension is the dimension of the annihilator \(A\) of \(N\). The case when \(N\) is graded and \(A\) is 1-dimensional is studied in detail, in particular the question to what extent the hypersurface \(S\) determines the algebra \(N\). A key example with nil-index 5 is given.

14J70 Hypersurfaces and algebraic geometry
32V40 Real submanifolds in complex manifolds
Full Text: DOI arXiv
[1] Cortiñas G., Krongold F.: Artinian algebras and differential forms. Commun. Algebra 27, 1711–1716 (1999) · Zbl 0931.13019 · doi:10.1080/00927879908826523
[2] Eastwood M.G.: Moduli of isolated hypersurface singularities. Asian J. Math. 8, 305–314 (2004) · Zbl 1084.32019 · doi:10.4310/AJM.2004.v8.n2.a6
[3] Elias, J., Rossi, M.E.: Isomorphism classes of short Gorenstein local rings via Macaulay’s inverse system. TAMS (2011, in press) · Zbl 1281.13015
[4] Fels G., Kaup W.: Local tube realizations of CR-manifolds and maximal abelian subalgebras. Ann. Sc. Norm. Sup. Pisa. Cl. Sci X, 99–128 (2011) · Zbl 1229.32020
[5] Fels, G., Kaup W.: Classification of commutative algebras and tube realizations of hyperquadrics. arXiv:0906.5549v2. (2011, to submit) · Zbl 1229.32020
[6] Fels G., Isaev A., Kaup W., Kruzhilin N.: Singularities and polynomial realizations of affine quadrics. J. Geom. Anal. 21, 767–782 (2011) · Zbl 1274.32018 · doi:10.1007/s12220-011-9223-y
[7] Isaev A.V.: On the number of affine equivalence classes of spherical tube hypersurfaces. Math. Ann. 349, 59–72 (2011) · Zbl 1207.32033 · doi:10.1007/s00208-010-0514-6
[8] Isaev, A.V.: On the affine homogeneity of algebraic hypersurfaces arising from Gorenstein algebras. Asian J. Math. http://arxiv.org/pdf/1101.0452v1 , see also page 3, line 11; http://arxiv.org/pdf/1101.0452v2 (2011, to appear)
[9] Mukai S.: An Introduction to Invariants and Moduli. Cambridge University Press, Cambridge (2003) · Zbl 1033.14008
[10] Perepechko, A.: On solvability of the automorphism group of a finite-dimensional algebra. arXiv:1012:0237 (2011, submitted) · Zbl 1301.13024
[11] Saito K.: Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14, 123–142 (1971) · Zbl 0224.32011 · doi:10.1007/BF01405360
[12] Xu Y.J., Yau S.S.T.: Micro-local characterization of quasi-homogeneous singularities. Am. J. Math. 118, 389–399 (1996) · Zbl 0927.32022 · doi:10.1353/ajm.1996.0020
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