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Extracting invariants of isolated hypersurface singularities from their moduli algebras. (English) Zbl 1277.32031
By a theorem due to J. N. Mather and S. S.-T. Yau [Invent. Math. 69, 243–251 (1982; Zbl 0499.32008)], two hypersurface germs in \((\mathbb C^m,0)\) with isolated singularities are biholomorphically equivalent if their moduli algebras are isomorphic. However, it is a very difficult problem to obtain the information of the singularity from its moduli algebra. In the paper under review, motivated by this problem, the authors consider the homogeneous singularities and construct invariants of graded Gorenstein \(\mathbb C\)-algebras of finite vector space dimension, using classical invariant theory. Their conjecture states that one can recover all absolute invariants of forms (homogeneous polynomials) of degree \(n\) from absolute invariants of forms of degree \(m(n-2)\) by evaluating so-called associated forms. They verify the conjecture for the cases \((m,n)=(3,3), (2,4), (2,5), (2,6)\), and provide the complete list of canonical forms with \((m,n)=(2,6)\).

MSC:
32S25 Complex surface and hypersurface singularities
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13A50 Actions of groups on commutative rings; invariant theory
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