Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture

The hypersurfaces of degree d in the projective space P-n correspond to points of p(N), where N = (d(n+d)) - 1. Now assume d = 2e is even, and let X-(n,X-d) subset of P-N denote the subvariety of two e-fold hyperplanes. We exhibit an upper bound on the Castelnuovo regularity of the ideal of X-(n,X-d), and show that this variety is r-normal for r >= 2. The latter result is representation-theoretic, and says that a certain GL(n+1)-equivariant morphism Sr(S-2e(Cn+1)) -> S-2 (S-re(Cn+1)) is surjective for r >= 2; a statement which is reminiscent of the Foulkes-Howe conjecture. For its proof, we reduce the statement to the case n = 1, and then show that certain transvectants of binary forms are nonzero. The latter part uses explicit calculations with Feynman diagrams and hypergeometric series. For ternary quartics and binary d-ics, we give explicit generators for the defining ideal of X-(n,X-d) expressed in the language of classical invariant theory. (c) 2006 Elsevier Inc. All rights reserved.