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Associated Forms: Current Progress and Open Problems

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Let \(d\ge 3\), \(n\ge 2\). The object of our study is the morphism \(\Phi \), introduced in earlier articles by J. Alper, M. Eastwood and the author, that assigns to every homogeneous form of degree d on \({\mathbb {C}}^n\) for which the discriminant \(\Delta \) does not vanish a form of degree \(n(d-2)\) on the dual space, called the associated form. This morphism is \({\mathrm{SL}}_n\)-equivariant and is of interest in connection with the well-known Mather–Yau theorem, specifically, with the problem of explicit reconstruction of an isolated hypersurface singularity from its Tjurina algebra. Letting p be the smallest integer such that the product \(\Delta ^p\Phi \) extends to the entire affine space of degree d forms, one observes that the extended map defines a contravariant. In the present paper, we survey known results on the morphism \(\Phi \), as well as the contravariant \(\Delta ^p\Phi \), and state several open problems. Our goal is to draw the attention of complex analysts and geometers to the concept of the associated form and the intriguing connection between complex singularity theory and invariant theory revealed through it.

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References

  1. Alper, J., Isaev, A.V.: Associated forms in classical invariant theory and their applications to hypersurface singularities. Math. Ann. 360, 799–823 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alper, J., Isaev, A.V.: Associated forms and hypersurface singularities: the binary case. J. Reine Angew. Math. https://doi.org/10.1515/crelle-2016-0008

  3. Alper, J., Isaev, A.V., Kruzhilin, N.G.: Associated forms of binary quartics and ternary cubics. Transform. Groups 21, 593–618 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benson, M.: Analytic equivalence of isolated hypersurface singularities defined by homogeneous polynomials. In: Singularities (Arcata, CA, 1981), Proceedings of Symposia in Pure Mathematics, vol. 40, pp. 111–118. American Mathematical Society, Providence, RI (1983)

  6. Bourbaki, N.: Commutative Algebra. Hermann, Paris (1972)

    MATH  Google Scholar 

  7. Cayley, A.: On the 34 concomitants of the ternary cubic. Am. J. Math. 4, 1–15 (1881)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dolgachev, I.: Classical Algebraic Geometry. A Modern View. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  9. Dolgachev, I.: Rational self-maps of moduli spaces, preprint, available from the Mathematics ArXiv at arXiv:1710.03373

  10. Eastwood, M.G.: Moduli of isolated hypersurface singularities. Asian J. Math. 8, 305–313 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eastwood, M.G., Isaev, A.V.: Extracting invariants of isolated hypersurface singularities from their moduli algebras. Math. Ann. 356, 73–98 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Elliott, E.B.: An Introduction to the Algebra of Quantics. Clarendon Press, Oxford (1895)

    Google Scholar 

  13. Emsalem, J.: Géométrie des points épais. Bull. Soc. Math. France 106, 399–416 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fedorchuk, M.: GIT semistability of Hilbert points of Milnor algebras. Math. Ann. 367, 441–460 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fedorchuk, M., Isaev, A.V.: Stability of associated forms. J. Algebraic Geom. ArXiv at arXiv:1703.00438

  16. Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants and Multidimensional Determinants. Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA (2008)

  17. Gordan, P.: Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Koeffizienten einer endlichen Anzahl solcher Formen ist. J. Reine Angew. Math. 69, 323–354 (1868)

    MathSciNet  MATH  Google Scholar 

  18. Görtz, U., Wedhorn, T.: Algebraic Geometry I. Schemes with Examples and Exercises, Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden (2010)

  19. Grace, J.H., Young, A.: The Algebra of Invariants. Reprint of the 1903 Original. Cambridge University Press, Cambridge (2010)

  20. Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Springer Monographs in Mathematics. Springer, Berlin (2007)

    MATH  Google Scholar 

  21. Greuel, G.-M., Pham, T.H.: Mather-Yau theorem in positive characterstic. J. Algebr. Geom. 26, 347–355 (2017)

    Article  MATH  Google Scholar 

  22. Harris, J.: Algebraic Geometry. A First Course. Graduate Texts in Mathematics, vol. 133. Springer, New York (1992)

    Google Scholar 

  23. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1997)

    Google Scholar 

  24. Hilbert, D.: Über die Theorie der algebraischen Formen. Math. Ann. 36, 473–534 (1890)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hilbert, D.: Über die vollen Invariantensysteme. Math. Ann. 42, 313–373 (1893)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hochschild, G.: The Structure of Lie Groups. Holden-Day, Inc., San Francisco (1965)

    MATH  Google Scholar 

  27. Huneke, C.: Hyman Bass and Ubiquity: Gorenstein Rings. In: Algebra, \(K\)-theory, Groups, and Education (New York, NY, 1997), Contemporary Mathematics, vol. 243, pp. 55–78. American Mathematical Society, Providence, RI (1999)

    Google Scholar 

  28. Isaev, A.V.: Application of classical invariant theory to biholomorphic classification of plane curve singularities, and associated binary forms. Proc. Steklov Inst. Math. 279, 245–256 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Isaev, A.V.: On two methods for reconstructing homogeneous hypersurface singularities from their Milnor algebras. Methods Appl. Anal. 21, 391–405 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Isaev, A.V.: On the contravariant of homogeneous forms arising from isolated hypersurface singularities. Int. J. Math. 27(11), 1650097 (14 pages, electronic) (2016). https://doi.org/10.1142/S0129167X1650097X

  31. Isaev, A.V.: A criterion for isomorphism of Artinian Gorenstein algebras. J. Commut. Alg. 8, 89–111 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Isaev, A.V.: On the image of the associated form morphism. In: Proceedings of the Japanese-Australian Workshop on Real and Complex Singularities (JARCS VI ) (Kagoshima, 2015 ), Saitama Math. J. 31, pp. 89–101 (2017)

  33. Isaev, A.V.: A combinatorial proof of the smoothness of catalecticant schemes associated to complete intersections. Ann. Combin. 21, 375–395 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Isaev, A.V., Kruzhilin, N.G.: Explicit reconstruction of homogeneous isolated hypersurface singularities from their Milnor algebras. Proc. Am. Math. Soc. 142, 581–590 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Iarrobino, A., Kanev, V.: Power Sums, Gorenstein Algebras, and Determinantal Loci. Lecture Notes in Mathematics, vol. 1721. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  36. Kung, J.P.S.: Canonical forms of binary forms: variations on a theme of Sylvester. In: Invariant Theory and Tableaux (Minneapolis, MN, 1988), IMA Vol. Math. Appl. 19. Springer, New York, pp. 46–58 (1990)

  37. Lakshmibai, V., Raghavan, K.: Standard Monomial Theory. Invariant Theoretic Approach. Invariant Theory and Algebraic Transformation Groups VIII, Encyclopaedia of Mathematical Sciences, vol. 137. Springer, Berlin (2008)

    MATH  Google Scholar 

  38. Mather, J., Yau, S.S.-T.: Classification of isolated hypersurface singularities by their moduli algebras. Invent. Math. 69, 243–251 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  39. Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  40. Mukai, S.: An Introduction to Invariants and Moduli. Cambridge Studies in Advanced Mathematics, vol. 81. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  41. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34. Springer, Berlin (1994)

    Google Scholar 

  42. Nagata, M.: Invariants of a group in an affine ring. J. Math. Kyoto Univ. 3, 369–377 (1963/1964)

  43. Newstead, P.E.: Introduction to Moduli Problems and Orbit Spaces. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 51, Tata Institute of Fundamental Research, Bombay. Narosa Publishing House, New Delhi (1978)

  44. Saito, K.: Einfach-elliptische Singularitäten. Invent. Math. 23, 289–325 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  45. Salmon, G.: A Treatise on the Higher Plane Curves: Intended as a Sequel to “A Treatise of Conic Sections”. Hodges, Foster and Co, Dublin (1879)

    MATH  Google Scholar 

  46. Scheja, G., Storch, U.: Über Spurfunktionen bei vollständigen Durchschnitten. J. Reine Angew. Math. 278/279, 174–190 (1975)

  47. Schmitt, A.H.W.: Geometric Invariant Theory and Decorated Principal Bundles. Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich (2008)

    Book  Google Scholar 

  48. Shoshitaĭshvili, A.N.: Functions with isomorphic Jacobian ideals. Funct. Anal. Appl. 10, 128–133 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  49. Sylvester, J.J.: A synoptical table of the irreducible invariants and covariants to a binary quintic, with a scholium on a theorem in conditional hyperdeterminants. Am. J. Math. 1, 370–378 (1878)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tauvel, P., Yu, R.W.T.: Lie Algebras and Algebraic Groups. Springer Monographs in Mathematics. Springer, Berlin (2005)

    Book  Google Scholar 

  51. Tevelev, E.A.: Projectively dual varieties. J. Math. Sci. (N.Y.) 117, 4585–4732 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  52. Vinberg, E.B., Onishchik, A.L.: Lie Groups and Algebraic Groups. Springer, Berlin (1990)

    MATH  Google Scholar 

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Acknowledgements

We are grateful to M. Fedorchuk for many very helpful discussions.

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  1. Mathematical Sciences Institute, Australian National University, Acton, Canberra, ACT, 2601, Australia

    Alexander Isaev

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Isaev, A. Associated Forms: Current Progress and Open Problems. J Geom Anal 29, 1706–1743 (2019). https://doi.org/10.1007/s12220-018-0058-7

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