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Associated forms in classical invariant theory and their applications to hypersurface singularities. (English) Zbl 1308.14048
Let \(Q^m_n\) be the vector space of forms of degree \(m\) on \(\mathbb{C}^n\). Let \(f\in Q^m_n\) with non–zero discriminant and \(M(f)\) the Milnor algebra of the isolated singularity at the origin of the hypersurface in \(\mathbb{C}^n\) defined by \(f\). The algebra \(M(f)\) gives rise to a form on \(\mathfrak{m}/\mathfrak{m}^2\) with values in the socle \(\text{Soc}(M(f))\) of \(M(f)\), \(\mathfrak{m}\) the maximal ideal of \(M(f)\). Using a canonical choice of coordinates in \(\mathfrak{m}/\mathfrak{m}^2\) and \(\text{Soc} (M(f))\) one obtains a form \(Q\) of degree \(n(m-2)\) on \(\mathbb{C}^n\). This form is called the associated form of \(f\).
It is proved that one can recover all absolute classical invariants of forms of degree \(m\) (i.e,. all \(\text{GL}(n, \mathbb{C})\) - invariant rational functions on \(Q^m_n\)) from those of forms of degree \(n(m-2)\) on \(\mathbb{C}^n\) by evaluating the latter on associated forms (a conjecture of M. G. Eastwood and A. V. Isaev [Math. Ann. 356, No. 1, 73–98 (2013; Zbl 1277.32031)]).

14L24 Geometric invariant theory
13A50 Actions of groups on commutative rings; invariant theory
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
32S25 Complex surface and hypersurface singularities
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