Level algebras with bad properties

This paper can be seen as a continuation of the works contained in the recent article (J. Alg., 305 (2006), 949-956) of the second author, and those of Juan Migliore (math. AC/0508067). Our results are:

1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for $e\gg 0$, we will construct a codimension three, type two $h$-vector of socle degree $e$ such that all the level algebras with that $h$-vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each $e\gg 0$.

2). There exist reduced level sets of points in ${\mathbf P}^3$ of type two whose artinian reductions all fail to have the WLP. Indeed, the examples constructed here have the same $h$-vectors we mentioned in 1).

3). For any integer $r\geq 3$, there exist non-unimodal monomial artinian level algebras of codimension $r$. As an immediate consequence of this result, we obtain another proof of the fact (first shown by Migliore in the above-mentioned preprint, Theorem 4.3) that, for any $r\geq 3$, there exist reduced level sets of points in ${\mathbf P}^r$ whose artinian reductions are non-unimodal.