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Invariants of Artinian Gorenstein Algebras and Isolated Hypersurface Singularities

  • Michael EastwoodEmail author
  • Alexander Isaev
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 38)

Abstract

We survey our recently proposed method for constructing biholomorphic invariants of quasihomogeneous isolated hypersurface singularities and, more generally, invariants of graded Artinian Gorenstein algebras. The method utilizes certain polynomials associated to such algebras, called nil-polynomials, and we compare them with two other classes of polynomials that have also been used to produce invariants.

Key words

Artinian Gorenstein algebras Isolated hypersurface singularities 

Mathematics Subject Classification (2010):

Primary 13H10 Secondary 13E10 32S25 13A50. 

References

  1. 1.
    Alper, J., Isaev, A., Associated forms in classical invariant theory and their applications to hypersurface singularities, Math. Ann., published online, DOI 10.1007/s00208-014-1054-2.Google Scholar
  2. 2.
    Bass, H., On the ubiquity of Gorenstein rings. Math. Z. 82 (1963), 8–28.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bruns, W., Herzog, J., Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge (1993).Google Scholar
  4. 4.
    Dieudonné, J. A., Carrell, J. B., Invariant theory, old and new. Adv. Math. 4 (1970), 1–80.CrossRefzbMATHGoogle Scholar
  5. 5.
    Eastwood, M. G., Moduli of isolated hypersurface singularities. Asian J. Math. 8 (2004), 305–313.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eastwood, M. G., Isaev, A. V., Extracting invariants of isolated hypersurface singularities from their moduli algebras. Math. Ann. 356 (2013), 73–98.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Elias, J., Rossi, M. E., Isomorphism classes of short Gorenstein local rings via Macaulay’s inverse system. Trans. Amer. Math. Soc. 364 (2012), 4589–4604.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Elliott, E. B., An Introduction to the Algebra of Quantics. Oxford University Press (1895).Google Scholar
  9. 9.
    Emsalem, J.: Géométrie des points épais. Bull. Soc. Math. France 106 (1978), 399–416.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fels, G., Isaev, A., Kaup, W., Kruzhilin, N., Isolated hypersurface singularities and special polynomial realizations of affine quadrics. J. Geom. Analysis 21 (2011), 767–782.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fels, G., Kaup, W., Nilpotent algebras and affinely homogeneous surfaces. Math. Ann. 353 (2012), 1315–1350MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gelfand, I. M., Kapranov, M. M., Zelevinsky, A. V., Discriminants, Resultants and Multidimensional Determinants. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA (2008).zbMATHGoogle Scholar
  13. 13.
    Greuel, G.-M., Lossen, C., Shustin, E., Introduction to Singularities and Deformations. Springer Monographs in Mathematics. Springer, Berlin (2007).zbMATHGoogle Scholar
  14. 14.
    Hertling, C., Frobenius Manifolds and Moduli Spaces for Singularities. Cambridge Tracts in Mathematics 151. Cambridge University Press, Cambridge (2002).Google Scholar
  15. 15.
    Hilbert, D., Ueber die Theorie der algebraischen Formen. Math. Ann. 36 (1890), 473–534.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Huneke, C., Hyman Bass and ubiquity: Gorenstein rings. In: Algebra, K-theory, Groups, and Education (New York, 1997). Contemp. Math. 243, pp. 55–78. Amer. Math. Soc., Providence, RI (1999).Google Scholar
  17. 17.
    Isaev, A. V., On the affine homogeneity of algebraic hypersurfaces arising from Gorenstein algebras. Asian J. Math. 15 (2011), 631–640.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mather, J., Yau, S. S.-T., Classification of isolated hypersurface singularities by their moduli algebras. Invent. Math. 69 (1982), 243–251.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mukai, S., An Introduction to Invariants and Moduli. Cambridge Studies in Advanced Mathematics 81. Cambridge University Press, Cambridge (2003).Google Scholar
  20. 20.
    Olver, P., Classical Invariant Theory. London Mathematical Society Student Texts 44. Cambridge University Press, Cambridge (1999).Google Scholar
  21. 21.
    Saito, K., Einfach-elliptische Singularitäten. Invent. Math. 23 (1974), 289–325.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sharpe, D. W., Vámos, P., Injective Modules. Cambridge Tracts in Mathematics and Mathematical Physics 62. Cambridge University Press, London–New York (1972).Google Scholar
  23. 23.
    Shoshitaishvili, A. N., Functions with isomorphic Jacobian ideals. Funct. Anal. Appl. 10 (1976), 128–133.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Mathematical Sciences InstituteThe Australian National UniversityCanberraAustralia

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