# zbMATH — the first resource for mathematics

Isolated hypersurface singularities and special polynomial realizations of affine quadrics. (English) Zbl 1274.32018
Let $$\mathcal O_n$$ be the ring of germs of holomorphic functions on $$(\mathbb C^n,o)$$ and let $$V=\{f=0\}\subset (\mathbb C^n,o)$$ with $$f\in \mathcal O_n$$. Then the quotient ring $$A(V):=\mathcal O_n/(f, \partial f/\partial z_1, \dots, \partial f/\partial z_n)$$ is called the moduli algebra of the germ of the hypersurface singularity $$(V,o)$$. By the theorem of J. N. Mather and S. S.-T. Yau [Invent. Math. 69, 243–251 (1982; Zbl 0499.32008)], two germs of isolated hypersurface singularities in $$(\mathbb C^n,o)$$ are biholomorphically equivalent if and only if their moduli algebras are isomorphic as $$\mathbb C$$-algebras. In the paper under review, the authors study the equivalence problem for admissible algebras, i.e., the maximal ideals of Gorenstein Artin algebras, and apply the results to the biholomorphic equivalence problem for quasi-homogeneous hypersurface singularities. In fact, the authors prove that it can be reduced to the linear equivalence problem for certain polynomials arising from the moduli algebras; furthermore these polynomials are completely determined by their quadratic and cubic terms. The last section includes concrete examples in which the equivalence problems in families of hypersurfaces are discussed.

##### MSC:
 32S25 Complex surface and hypersurface singularities 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 14B05 Singularities in algebraic geometry
Full Text:
##### References:
 [1] Aramova, A.: Graded analytic algebras over a field of positive characteristic. Math. Z. 215, 631–640 (1994) · Zbl 0790.13006 · doi:10.1007/BF02571734 [2] Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963) · Zbl 0112.26604 · doi:10.1007/BF01112819 [3] Chen, H., Seeley, C., Yau, S.S.-T.: Algebraic determination of isomorphism classes of the moduli algebras of $$$\backslash$$tilde{E}_{6}$ singularities. Math. Ann. 318, 637–666 (2000) · Zbl 0976.32015 · doi:10.1007/s002080000132 [4] Cortiñas, G., Krongold, F.: Artinian algebras and differential forms. Commun. Algebra 27, 1711–1716 (1999) · Zbl 0931.13019 · doi:10.1080/00927879908826523 [5] Eastwood, M.G.: Moduli of isolated hypersurface singularities. Asian J. Math. 8, 305–313 (2004) · Zbl 1084.32019 · doi:10.4310/AJM.2004.v8.n2.a6 [6] Eastwood, M.G., Ezhov, V.V.: On affine normal forms and a classification of homogeneous surfaces in affine three-space. Geom. Dedic. 77, 11–69 (1999) · Zbl 0999.53008 · doi:10.1023/A:1005083518793 [7] Eisenbud, D.: Commutative Algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995) · Zbl 0819.13001 [8] Ezhov, V., Schmalz, G.: A matrix Poincaré formula for holomorphic automorphisms of quadrics of higher codimension. Real associative quadrics. J. Geom. Anal. 8, 27–41 (1998) · Zbl 0957.32016 · doi:10.1007/BF02922107 [9] Fels, G., Kaup, W.: Local tube realizations of CR-manifolds and maximal abelian subalgebras. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) X, 99–128 (2011) · Zbl 1229.32020 [10] Fels, G., Kaup, W.: Classification of commutative algebras and tube realizations of hyperquadrics. Preprint, available from http://arxiv.org/pdf/0906.5549v2 · Zbl 1229.32020 [11] Fels, G., Kaup, W.: Nilpotent algebras and affinely homogeneous surfaces. Preprint, available from http://arxiv.org/pdf/1101.3088v2 · Zbl 1254.14053 [12] Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Springer Monographs in Mathematics. Springer, Berlin (2007) · Zbl 1125.32013 [13] Hertling, C.: Generic Torelli for semiquasihomogeneous singularities. In: Trends in Singularities. Trends Math., pp. 115–140. Birkhäuser, Basel (2002) · Zbl 1025.32025 [14] Hertling, C.: Frobenius Manifolds and Moduli Spaces for Singularities. Cambridge Tracts in Mathematics, vol. 151. Cambridge University Press, Cambridge (2002) · Zbl 1023.14018 [15] Huneke, C.: Hyman Bass and ubiquity: Gorenstein rings. In: Algebra, K-theory, Groups, and Education, New York, 1997. Contemp. Math., vol. 243, pp. 55–78. Amer. Math. Soc., Providence (1999) · Zbl 0960.13008 [16] Kunz, E.: Almost complete intersections are not Gorenstein rings. J. Algebra 28, 111–115 (1974) · Zbl 0275.13025 · doi:10.1016/0021-8693(74)90025-8 [17] Leichtweiß, K.: Über eine geometrische Deutung des Affinnormalenvektors einseitig gekrümmter Hyperflächen. Arch. Math. 53, 613–621 (1989) · Zbl 0661.53005 · doi:10.1007/BF01199822 [18] Mather, J., Yau, S.S.-T.: Classification of isolated hypersurface singularities by their moduli algebras. Invent. Math. 69, 243–251 (1982) · Zbl 0499.32008 · doi:10.1007/BF01399504 [19] Saito, K.: Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14, 123–142 (1971) · Zbl 0224.32011 · doi:10.1007/BF01405360 [20] Saito, K.: Einfach-elliptische Singularitäten. Invent. Math. 23, 289–325 (1974) · Zbl 0296.14019 · doi:10.1007/BF01389749 [21] Seeley, C., Yau, S.S.-T.: Variation of complex structures and variation of Lie algebras. Invent. Math. 99, 545–565 (1990) · Zbl 0685.14004 · doi:10.1007/BF01234430 [22] 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
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.