On equations defining Coincident Root loci

Abstract. Given integers n, d, e with 1 ≤ e < d 2 , let X ⊆ P ( d+n d )−1 denote the locus of degree d hypersurfaces in P n which are supported on two hyperplanes with multiplicities d − e and e. Thus X is the BrillGordan locus associated to the partition (d − e, e). The main result of the paper is an exact determination of the Castelnuovo regularity of the ideal of X. Moreover we show that X is r-normal for r ≥ 3.In the case of binary forms (i.e., for n = 1) we give an invariant theoretic description of the ideal generators, and furthermore exhibit a set of two covariants which define this locus set-theoretically.In addition to the standard cohomological tools in algebraic geometry, the proof crucially relies on the nonvanishing of certain 3j-symbols from the quantum theory of angular momentum.

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“…Together with [1, Theorem 1.4], this completely proves the following result which was first conjectured in [7]. Theorem 1.3.…”

Section: Binary Formssupporting

confidence: 79%

The bipartite Brill-Gordan locus and angular momentum

Abdesselam

Chipalkatti

2006

Transformation Groups

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“…well-known book by Gelfand et al [7]) but an alternative term ''generalised coincident root loci'' looks too long and not much better. We would like to mention that the problem of finding the algebraic equations defining the strata in the discriminants is nontrivial and goes back to Arthur Cayley [2] (see [4,9,27] for the recent results in this direction).…”

Section: Generalised Discriminants and Deformed Cms Operatorsmentioning

confidence: 99%

Generalised discriminants, deformed Calogero–Moser–Sutherland operators and super-Jack polynomials

Сергеев

Веселов

2005

Advances in Mathematics

158

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It is shown that the deformed Calogero-Moser-Sutherland (CMS) operators can be described as the restrictions on certain affine subvarieties (called generalised discriminants) of the usual CMS operators for infinite number of particles. The ideals of these varieties are shown to be generated by the Jack symmetric functions related to the Young diagrams with special geometry. A general structure of the ideals which are invariant under the action of the quantum CMS integrals is discussed in this context. The shifted super-Jack polynomials are introduced and combinatorial formulas for them and for super-Jack polynomials are given. r 2004 Elsevier Inc. All rights reserved.

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“…Secondly, (and what is more to the point) I believe that the construction used here is of interest not confined to this example. In [4], it was used on varieties defined by binary forms having roots of specified multiplicities. The symmetric group S 3 acts on (PV * ) 3 by permuting the factors, and X ∆ is the categorial quotient (PV * ) 3 /S 3 .…”

Section: 5mentioning

confidence: 99%

“…Hence there is one concomitant each of type (8, 4, 1), (8,1,4). Now Φ 841 is necessarily equal to the product Φ 400 Φ 441 .…”

Section: ])mentioning

confidence: 99%