Contents
Part 1. Nonexistence and Existence 1
Chapter 1. Introduction 3
Chapter 2. Numerical Conditions 7
Chapter 3. Homological Methods 13
Chapter 4. Some Refinements 27
Chapter 5. Constructing Artinian Level Algebras 31
5.1. Inverse Systems 31
5.2. Level Quotients of the Co-ordinate Rings of Points 37
5.3. Constructing Level Algebras with the WLP 40
5.4. The Linked-Sum Method 48
Chapter 6. Constructing Level Sets of Points 55
6.1. Method 1: Intersection of Suitable Curves 55
6.2. Method 2: Liaison Tricks 58
6.3. Method 3: Union of suitable sets of points 62
6.4. Method 4: General Sets of Points on Suitable curves 64
Chapter 7. Expected Behavior 67
Part 2. Appendix: A Classification of Codimension Three Level
Algebras of Low Socle Degree 73
Appendix A. Introduction and Notation 75
Appendix B. Socle Degree 6 and Type 2 77
Appendix C. Socle Degree 5 93
Appendix D. Socle Degree 4 119
Appendix E. Socle Degree 3 129
Appendix F. Summary 133
Appendix. Bibliography 137
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