Introduction to Liaison Theory and Deficiency Modules

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August 1998



In the fall of 1992 I was invited by Professor Changho Keem to visit Seoul National University and give a series of talks. I was asked to write a monograph based on my talks, and the result was published by the Global Analysis Research Center of that University in 1994. The monograph treated deficiency modules and liaison theory for complete intersections. Over the next several years I continually thought of improvements and additions that I would like to make to the manuscript, and at the same time my research led me in directions that gave me a fresh perspective on much of the material, especially in the direction of liaison theory. This re­ sulted in a dramatic change in the focus of this manuscript, from complete intersections to Gorenstein ideals, and a substantial amount of additions and revisions. It is my hope that this book now serves not only as an introduction to a beautiful subject, but also gives the reader a glimpse at very recent developments and an idea of the direction in which liaison theory is going, at least from my perspective. One theme which I have tried to stress is the tremendous amount of geometry which lies at the heart of the subject, and the beautiful interplay between algebra and geometry. Whenever possible I have given remarks and examples to illustrate this interplay, and I have tried to phrase the results in as geometric a way as possible.


1 Background.- 1.1 Finitely Generated Graded S-Modules.- 1.2 The Deficiency Modules (Mi)(V).- 1.3 Hyperplane and Hypersurface Sections.- 1.4 Artinian Reductions and h-Vectors.- 1.5 Examples.- 2 Submodules of the Deficiency Module.- 2.1 Measuring Deficiency.- 2.2 Generalizing Dubreil's Theorem.- 2.3 Lifting the Cohen-Macaulay Property.- 3 Buchsbaum Curves and Liaison Addition.- 3.1 Buchsbaum Curves.- 3.2 Liaison Addition.- 3.3 Constructing Buchsbaum Curves in P3.- 4 Gorenstein Subschemes of Projective Space.- 4.1 Basic Results on Gorenstein Ideals.- 4.2 Constructions of Gorenstein Schemes.- 4.2.1 Intersection of Linked Schemes.- 4.2.2 Sections of Buchsbaum-Rim Sheaves of Odd Rank.- 4.2.3 Linear Systems on aCM Schemes.- 4.3 Codimension Three Gorenstein Ideals.- 5 Liaison Theory in Arbitrary Codimension.- 5.1 Definitions and First Examples.- 5.2 Relations Between Linked Schemes.- 5.3 The Hartshorne-Schenzel Theorem.- 5.4 The Structure of an Even Liaison Class.- 5.5 Geometric Invariants of a Liaison Class.- 6 Liaison Theory in Codimension Two.- 6.1 The aCM Situation and Generalizations.- 6.2 Rao's Results.- 6.3 The Lazarsfeld-Rao Property.- 6.4 Applications.- 6.4.1 Smooth Curves in P3.- 6.4.2 Smooth Surfaces in IP4 and Threefolds in P5.- 6.4.3 Possible Degrees and Genera in a Codimension Two Even Liaison Class.- 6.4.4 Stick Figures.- 6.4.5 Low Rank Vector Bundles and Schemes Defined by a Small Number of Equations.


"Suitable for a graduate course in algebraic geometry...numerous discussions about the historical development, ample and useful list of references to important original sources... Relations between linked schemes...are explained in great detail, with complete proofs and numerous examples. ...The book ends with a section which gives a flavour of some of the ways in which liaison theory has been applied in the literature. Many more applications and examples are spread throughout the text. They contribute to a lively and inspiring style. This book is a worthwhile addition to every algebraic geometer's library."
--Mathematical Reviews
"A highly specialized monograph that provides a very good introduction to contemporary research in the fields of liaison theory and deficiency modules... The author pays great attention to motivation and the geometric aspects of the theory. There are many examples through which the reader is introduced into the theory, thereby stimulating research in the field... Useful both for specialists and for postgraduate students."
--EMS Newsletter
EAN: 9780817640279
ISBN: 0817640274
Untertitel: 'Progress in Mathematics'. 2nd ed. 1998. Book. Sprache: Englisch.
Verlag: Birkhäuser
Erscheinungsdatum: August 1998
Seitenanzahl: 236 Seiten
Format: gebunden
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