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Extracting invariants of isolated hypersurface singularities from their moduli algebras

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Abstract

We use classical invariant theory to construct invariants of complex graded Gorenstein algebras of finite vector space dimension. As a consequence, we obtain a way of extracting certain numerical invariants of quasi-homogeneous isolated hypersurface singularities from their moduli algebras, which extends an earlier result due to the first author. Furthermore, we conjecture that the invariants so constructed solve the biholomorphic equivalence problem in the homogeneous case. The conjecture is easily verified for binary quartics and ternary cubics. We show that it also holds for binary quintics and sextics. In the latter cases the proofs are much more involved. In particular, we provide a complete list of canonical forms of binary sextics, which is a result of independent interest.

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Notes

  1. The formulae for the discriminant that we use below in the cases \(m=2,3\) differ from the one given in [8] by scalar factors.

  2. This proof was suggested to us by A. Gorinov.

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Acknowledgments

We would like to thank A. Gorinov for suggesting a proof of Proposition 3.1. This work is supported by the Australian Research Council.

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Authors and Affiliations

  1. Mathematical Sciences Institute, The Australian National University, Canberra, ACT, 0200, Australia

    M. G. Eastwood & A. V. Isaev

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  1. M. G. Eastwood
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  2. A. V. Isaev
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Correspondence to A. V. Isaev.

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Eastwood, M.G., Isaev, A.V. Extracting invariants of isolated hypersurface singularities from their moduli algebras. Math. Ann. 356, 73–98 (2013). https://doi.org/10.1007/s00208-012-0836-7

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  • DOI: https://doi.org/10.1007/s00208-012-0836-7

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