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 [35], [32], [55],

[54], [71]). 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

n

and also explain some results of Boij [8] 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 [23] 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

situations.

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 .