We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree zero, we describe the supporting hyperplanes and extreme rays for the cones generate
d by the Hilbert functions of all modules, all modules with bounded a-invariant, and all modules with bounded Castelnuovo-Mumford regularity. The first of these cones is infinite-dimensional and simplicial, the second is finite-dimensional but neither simplicial nor polyhedral, and the third is finite-dimensional and simplicial.
We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the
first open case, namely Hilbert function (1,3,6,6,3,1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from $mathbb P^2$ to $mathbb P^3$, and Hesse configurations in $mathbb P^2$.
We generalize an example, due to Sylvester, and prove that any monomial of degree $d$ in $mathbb R[x_0, x_1]$, which is not a power of a variable, cannot be written as a linear combination of fewer than $d$ powers of linear forms.
