Associated Forms and Hypersurface Singularities: the Binary Case
نویسنده
چکیده
In the recent articles [EI] and [AI], it was conjectured that all rational GLn-invariant functions of forms of degree d ≥ 3 on Cn can be extracted, in a canonical way, from those of forms of degree n(d − 2) by means of assigning to every form with nonvanishing discriminant the so-called associated form. The conjecture was confirmed in [EI] for binary forms of degree d ≤ 6 as well as for ternary cubics. Furthermore, a weaker version of it was settled in [AI] for arbitrary n and d. In the present paper, we focus on the case n = 2 and establish the conjecture, in a rather explicit way, for binary forms of an arbitrary degree.
منابع مشابه
Examples of Application of Nil-polynomials to the Biholomorphic Equivalence Problem for Isolated Hypersurface Singularities
Let V1, V2 be hypersurface germs in C m withm ≥ 2, each having a quasi-homogeneous isolated singularity at the origin. In our recent article [7] we reduced the biholomorphic equivalence problem for V1, V2 to verifying whether certain polynomials, called nilpolynomials, that arise from the moduli algebras of V1, V2 are equivalent up to scale by means of a linear transformation. In this paper we ...
متن کاملAssociated Forms in Classical Invariant Theory and Their Applications to Hypersurface Singularities
It was conjectured in the recent article [EI] that all absolute classical invariants of forms of degree m ≥ 3 on C can be extracted, in a canonical way, from those of forms of degree n(m−2) by means of assigning every form with non-vanishing discriminant the so-called associated form. This surprising conjecture was confirmed in [EI] for binary forms of degree m ≤ 6 and ternary cubics. In the pr...
متن کاملRings of Singularities
This paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``Advanced School and Conference on homological and geometrical methods in representation theory'' at the International Centre for Theoretical Physics (ICTP), Miramare - Trieste, Italy, 11-29 January 2010. We show how to associate to a triple of posit...
متن کاملStability of Associated Forms
We show that the associated form, or, equivalently, a Macaulay inverse system, of an Artinian complete intersection of type (d, . . . , d) is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities.
متن کاملResidue forms on singular hypersurface
Let M be a complex manifold, X ⊂ M a singular hypersurface. We study residues of top-dimensional forms with poles along X. Applying resolution of singularities we obtain residue classes either in L-cohomology of X or in the intersection cohomology. The conditions allowing to construct these residue classes coincide. They can be formulated in terms of weight filtration.
متن کامل