  In a recent paper, F. Zanello showed that level Artinian algebras in 3
variables can fail to have the Weak Lefschetz Property (WLP), and can even fail
to have unimodal Hilbert function. We show that the same is true for the
Artinian reduction of reduced, level sets of points in projective 3-space. Our
main goal is to begin an understanding of how the geometry of a set of points
can prevent its Artinian reduction from having WLP, which in itself is a very
algebraic notion. More precisely, we produce level sets of points whose
Artinian reductions have socle types 3 and 4 and arbitrary socle degree $\geq
12$ (in the worst case), but fail to have WLP. We also produce a level set of
points whose Artinian reduction fails to have unimodal Hilbert function; our
example is based on Zanello's example. Finally, we show that a level set of
points can have Artinian reduction that has WLP but fails to have the Strong
Lefschetz Property. While our constructions are all based on basic double
G-linkage, the implementations use very different methods.
