Introduction to Liaison Theory and Deficiency Modules

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.