9 D ec 2 00 5 BAD PROPERTIES OF LEVEL ALGEBRAS

This paper can be seen as a continuation of the works contained in the recent preprints [Za], of the second author, and [Mi], of Juan Migliore. 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 ≫ 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 ≫ 0. 2). There exist reduced level sets of points in 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 ≥ 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 [Mi, Theorem 4.3]) that, for any r ≥ 3, there exist reduced level sets of points in P r whose artinian reductions are non-unimodal.