  Let n,d be positive integers, with d even (say d=2e). Let X_(n,d) denote the
locus of degree d hypersurfaces in P^n which consist of two e-fold hyperplanes.
We bound the regularity of the ideal of this variety. Moreover, we show that
this variety is r-normal for r at least 2. The proof of the latter part is is a
result of a tripartite collaboration of algebraic geometry, classical invariant
theory and theoretical physics. It is executed by reducing the question to a
combinatorial calculation involving Feynman diagrams and hypergeometric
functions.
