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

The hypersurfaces of degree d   in the projective space PnPn correspond to points of PNPN, where N=(n+dd)−1. Now assume d=2ed=2e is even, and let X(n,d)⊆PNX(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)X(n,d), and show that this variety is r  -normal for r⩾2r⩾2. The latter result is representation-theoretic, and says that a certain GLn+1GLn+1-equivariant morphismSr(S2e(Cn+1))→S2(Sre(Cn+1))Sr(S2e(Cn+1))→S2(Sre(Cn+1)) is surjective for r⩾2r⩾2; a statement which is reminiscent of the Foulkes–Howe conjecture. For its proof, we reduce the statement to the case n=1n=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)X(n,d) expressed in the language of classical invariant theory.