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

AbstractThe hypersurfaces of degree d in the projective space Pn correspond to points of PN, where N=(n+dd)−1. Now assume d=2e is even, and let X(n,d)⊆PN denote the subvariety of two e-fold hyperplanes. We exhibit an upper bound on the Castelnuovo regularity of the ideal of X(n,d), and show that this variety is r-normal for r⩾2. The latter result is representation-theoretic, and says that a certain GLn+1-equivariant morphismSr(S2e(Cn+1))→S2(Sre(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,d) expressed in the language of classical invariant theory