Brill-Gordan Loci, Transvectants and an Analogue of the Foulkes Conjecture

Combining a selection of tools from modern algebraic geometry, representation theory, the classical invariant theory of binary forms, together with explicit calculations with hypergeometric series and Feynman diagrams, we obtain the following interrelated results. A Castelnuovo-Mumford regularity bound and a projective normality result for the locus of hypersufaces that are equally supported on two hyperplanes. The surjectivity of an equivariant map between two plethystic compositions of symmetric powers; a statement which is reminiscent of the Foulkes-Howe conjecture. The nonvanishing of even transvectants of exact powers of generic binary forms. The nonvanishing of a collection of symmetric functions defined by sums over magic squares and transportation matrices with nonnegative integer entries. An explicit set of generators, in degree three, for the ideal of the coincident root locus of binary forms with only two roots of equal multiplicity.Comment: This is a considerably expanded version of math.AG/040523

The slopes determined by n points in the plane

Let $m_{12}$, $m_{13}$, ..., $m_{n-1,n}$ be the slopes of the $\binom{n}{2}$ lines connecting $n$ points in general position in the plane. The ideal $I_n$ of all algebraic relations among the $m_{ij}$ defines a configuration space called the {\em slope variety of the complete graph}. We prove that $I_n$ is reduced and Cohen-Macaulay, give an explicit Gr\"obner basis for it, and compute its Hilbert series combinatorially. We proceed chiefly by studying the associated Stanley-Reisner simplicial complex, which has an intricate recursive structure. In addition, we are able to answer many questions about the geometry of the slope variety by translating them into purely combinatorial problems concerning enumeration of trees.Comment: 36 pages; final published versio