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On the image of the associated form morphism. (English) Zbl 1378.14044
Summary: Let $$\mathbb{C}[x_1,\dots, x_n]_{d+1}$$ be the vector space of homogeneous forms of degree $$d+1$$ on $$\mathbb{C}^n$$, with $$n$$, $$d\geq 2$$. In earlier articles by M. G. Eastwood and the author [Math. Ann. 356, No. 1, 73–98 (2013; Zbl 1277.32031)], J. Alper and the author [Math. Ann. 360, No. 3–4, 799–823 (2014; Zbl 1308.14048); “Associated forms and hypersurface singularities: The binary case”, to appear in J. reine angew. Math., published online, doi:10.1515/crelle-2016-0008)], J. Alper et al. [Transform. Groups 21, No. 3, 593–618 (2016; Zbl 1372.14038)], we introduced a morphism, called $$A$$, that assigns to every nondegenerate form the so-called associated form lying in the space $$\mathbb{C}[y_1,\dots, y_n]_{n(d-1)}$$.
One of the reasons for our interest in $$A$$ is the conjecture – motivated by the well-known Mather-Yau theorem on complex isolated hypersurface singularities asserting that all regular $$\text{GL}_n$$-invariant functions on the affine open subvariety $$\mathbb{C}[x_1,\dots,x_n]_{d+1,\Delta}$$ of forms with nonvanishing discriminant can be obtained as the pull-backs by means of $$A$$ of the rational $$\text{GL}_n$$-invariant functions on $$\mathbb{C}[y_1,\dots, y_n]_{n(d-1)}$$ defined on im$$(A)$$. The morphism $$A$$ factors as $$A={\mathbf A}\circ\text{grad}$$, where grad is the gradient morphism and $${\mathbf A}$$ assigns to every $$n$$-tuple of forms of degree $$d$$ with nonvanishing resultant a form in $$\mathbb{C}[y_1,\dots, y_n]_{n(d-1)}$$ defined analogously to $$A(f)$$ for a nondegenerate $$f$$. In order to establish the conjecture, it is important to study the image of $${\mathbf A}$$.
In the present paper, we show that im$$({\mathbf A})$$ is an open subset of an irreducible component of each of the so-called catalecticant varieties $$V$$, Gor$$(T)$$ and describe the closed complement to im$$({\mathbf A})$$, at the same time clarifying and extending known results on these varieties.
Furthermore, for $$n=3$$, $$d=2$$ we give a description of the complement to im$$({\mathbf A})$$ via the zero locus of the Aronhold invariant of degree 4, which establishes an analogy with the case $$n=2$$ where this complement is known to be the vanishing locus of the catalecticant for any $$d\geq 2$$.

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