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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)]).

##### MSC:
 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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