On the image of the associated form morphism.

*(English)*Zbl 1378.14044Summary: 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 \(d\geq 2\).

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 |