1. INTRODUCTION 5
Stanley as the "godfather" of this concept. The study of this property has also
been taken up by several other authors for Gorenstein algebras (see , , ,
, ). Ours is the first systematic discussion of the WLP for level algebras
(see Propositions 5.11, 5.15, 5.16, 5.18, 5.24, Corollary 5.17, and Example 6.18).
Chapters 5 and 6 are devoted to construction methods for level Artin algebras,
reduced level algebras of positive Krull dimension and level algebras with the WLP.
In section 5.1 we concentrate on the construction of Artin level algebras using
Inverse Systems. In subsection 5.2 we explore level quotients of co-ordinate rings
of sets of points in P
and also explain some results of Boij  in this direction.
In subsection 5.3 we concentrate on constructing level algebras which have the
WLP. In subsection 5.4 we explain the "linked-sum" method for constructing level
Artinian algebras. Since the "linked-sum" method requires us to have level algebras
of positive Krull dimension readily at hand, we recall some results from  which
explain how to construct (easily) useful sets of level points. Although the "linked-
sum" method is very powerful for constructing level algebras we show (Remark 5.32)
that it is not always possible to use it.
In Chapter 6 we consider the problem of constructing level sets of points in a
more general way than we had considered earlier. We give four, essentially different,
construction methods. Each of these methods is used to construct new examples
of level algebras.
Chapter 7 is more speculative. There are natural candidates for level algebras,
both at the Artinian level and at the points level, obtained by making "general"
choices. We give a preliminary result and a conjecture, respectively, for these two
In a (rather large) Appendix we give a complete list of the h-vectors of level
Artin algebras of codimension 3 having socle degree 5 and of codimension 3,
socle degree 6 and type 2. In all these cases we show that for each h-vector in our
lists there is an example of a level algebra with that h-vector and having the WLP.
For socle degree 4 and for type 2 in socle degrees 5 and 6 we show that every
h-vector in our list is also the h-vector of a level set of points in P3.
We would like to take this opportunity to thank A. Iarrobino for his interest and
support for this project and for generously sharing his insights about level algebras
with us. We also would like to thank R. Stanley and B. Ulrich for an interesting
discussion about our Theorem 2.10, and G. Dalzotto for the CoCoA program that
was used to generate some of the initial lists in the appendix. It is also a pleasure
to thank the MSRI for its kind hospitality to the first author during part of the
writing of this monograph.
The authors dedicate their work on this book as follows: Geramita to his newly
arrived (Dec. 2004) and much awaited first grandchild, Sophia Clara; Harima to his
late father, Isamu Harima, who courageously endured his illness and died during
the writing of this book; Migliore to his beloved parents, Maria Teresa Migliore and
the late Francisco Migliore; and Shin to the memory of his late father, Sung-Ho
Shin, and to his mother, Kyoung-Rye Kang, who has been fighting her illness in a
hospital for a few years .