Introduction to Liaison Theory and Deficiency Modules

Part 1 Background: finitely generated graded S-modules; the deficiency modules (Mi)(V); hyperplane and hypersurface sections; Artinian reductions and h-vectors; examples. Part 2 Submodules of the deficiency module: measuring deficiency; generalizing Dubreil's theorem; lifting the Cohen-Macaulay property. Part 3 Buchsbaum curves: liaison addition; constructing Buchsbaum curves in P3. Part 4 Gorenstein subschemes of Pn: basic results on Gorenstein ideals; constructions of Gorenstein schemes - intersection of linked schemes, sections of Buchsbaum-Rim sheaves of odd rank, linear systems on aCM scheme; codimension three Gorenstein ideals. Part 5 Liaison theory: definition and first example; relations between linked schemes; the Hartshorne-Schenzel theorem; the structure of an even liaison class; geometric invariants of a liaison class. Part 6 Liaison theory in codimension two: the aCM situation and generalizations; Rao's results; the Lazarsfeld-Rao property; applications - smooth curves in P3, smooth surfaces in P4 and threefold in P5, possible degrees and genera in a codimension two even liaison class, stick figures, low rank vector bundles and schemes defined by a small number of equations.