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Associated forms: current progress and open problems. (English) Zbl 1468.13008

Author’s abstract: Let \(d\ge 3\), \(n\ge 2\). The object of our study is the morphism \(\Phi \), introduced in earlier articles by J. Alper and the author [Math. Ann. 360, No. 3–4, 799–823 (2014; Zbl 1308.14048); J. Reine Angew. Math. 745, 83–104 (2018; Zbl 1403.14060)]; M. G. Eastwood [Asian J. Math. 8, No. 2, 305–314 (2004; Zbl 1084.32019)] and M. G. Eastwood and the author [Math. Ann. 356, No. 1, 73–98 (2013; Zbl 1277.32031)], 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.

MSC:

13A50 Actions of groups on commutative rings; invariant theory
14L24 Geometric invariant theory
32S25 Complex surface and hypersurface singularities
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