×

zbMATH — the first resource for mathematics

Explicit reconstruction of homogeneous isolated hypersurface singularities from their Milnor algebras. (English) Zbl 1305.32016
For any complex analytic isolated hypersurface singularity in \(({\mathbb C}^n,0)\) one can consider its moduli (or Tjurina) algebra. By a theorem due to J. N. Mather and S. S. T. Yau [Invent. Math. 69, 243–251 (1982; Zbl 0499.32008)], for any fixed \(n\), two isolated hypersurface singularities are biholomorphically equivalent if and only if their moduli algebras are isomorphic. The proof of the Mather-Yau theorem does not provide any explicit procedure for the reconstruction of the germs from their moduli algebras. The present article provides such an explicit method for the construction of the singular germ whenever the singularity is homogeneous. In this case the moduli algebra equals the Milnor algebra, and the dimension \(n\) also can be recovered from the algebra as its embedding dimension. The method is illustrated on a concrete example.

MSC:
32S25 Complex surface and hypersurface singularities
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8 – 28. · Zbl 0112.26604
[2] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. · Zbl 0788.13005
[3] Michael G. Eastwood, Moduli of isolated hypersurface singularities, Asian J. Math. 8 (2004), no. 2, 305 – 313. · Zbl 1084.32019
[4] M. G. Eastwood and A. V. Isaev, Extracting invariants of isolated hypersurface singularities from their moduli algebras, Math. Ann. 356 (2013), no. 1, 73 – 98. · Zbl 1277.32031
[5] E. B. Elliott, An introduction to the algebra of quantics, Oxford University Press, 1895.
[6] G. Fels, A. Isaev, W. Kaup, and N. Kruzhilin, Isolated hypersurface singularities and special polynomial realizations of affine quadrics, J. Geom. Anal. 21 (2011), no. 3, 767 – 782. · Zbl 1274.32018
[7] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008. Reprint of the 1994 edition. · Zbl 1138.14001
[8] G.-M. Greuel, C. Lossen, and E. Shustin, Introduction to singularities and deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007. · Zbl 1125.32013
[9] Craig Huneke, Hyman Bass and ubiquity: Gorenstein rings, Algebra, \?-theory, groups, and education (New York, 1997) Contemp. Math., vol. 243, Amer. Math. Soc., Providence, RI, 1999, pp. 55 – 78. · Zbl 0960.13008
[10] John N. Mather and Stephen S. T. Yau, Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math. 69 (1982), no. 2, 243 – 251. · Zbl 0499.32008
[11] Kyoji Saito, Einfach-elliptische Singularitäten, Invent. Math. 23 (1974), 289 – 325 (German). · Zbl 0296.14019
[12] Mathias Schulze, A solvability criterion for the Lie algebra of derivations of a fat point, J. Algebra 323 (2010), no. 10, 2916 – 2921. · Zbl 1204.17014
[13] Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57 – 83. · Zbl 0384.13012
[14] Stephen S.-T. Yau, Solvable Lie algebras and generalized Cartan matrices arising from isolated singularities, Math. Z. 191 (1986), no. 4, 489 – 506. · Zbl 0589.17012
[15] Stephen S.-T. Yau, Solvability of Lie algebras arising from isolated singularities and nonisolatedness of singularities defined by \?\?(2,\?) invariant polynomials, Amer. J. Math. 113 (1991), no. 5, 773 – 778. · Zbl 0747.17012
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.