Memoirs of the American Mathematical Society 2001; 116 pp; softcover Volume: 154 ISBN10: 0821827383 ISBN13: 9780821827383 List Price: US$52 Individual Members: US$31.20 Institutional Members: US$41.60 Order Code: MEMO/154/732
 This paper contributes to the liaison and obstruction theory of subschemes in \(\mathbb{P}^n\) having codimension at least three. The first part establishes several basic results on Gorenstein liaison. A classical result of Gaeta on liaison classes of projectively normal curves in \(\mathbb{P}^3\) is generalized to the statement that every codimension \(c\) "standard determinantal scheme" (i.e. a scheme defined by the maximal minors of a \(t\times (t+c1)\) homogeneous matrix), is in the Gorenstein liaison class of a complete intersection. Then Gorenstein liaison (Gliaison) theory is developed as a theory of generalized divisors on arithmetically CohenMacaulay schemes. In particular, a rather general construction of basic double Glinkage is introduced, which preserves the even Gliaison class. This construction extends the notion of basic double linkage, which plays a fundamental role in the codimension two situation. The second part of the paper studies groups which are invariant under complete intersection linkage, and gives a number of geometric applications of these invariants. Several differences between Gorenstein and complete intersection liaison are highlighted. For example, it turns out that linearly equivalent divisors on a smooth arithmetically CohenMacaulay subscheme belong, in general, to different complete intersection liaison classes, but they are always contained in the same even Gorenstein liaison class. The third part develops the interplay between liaison theory and obstruction theory and includes dimension estimates of various Hilbert schemes. For example, it is shown that most standard determinantal subschemes of codimension \(3\) are unobstructed, and the dimensions of their components in the corresponding Hilbert schemes are computed. Readership Graduate students and research mathematicians interested in algebraic geometry. Table of Contents  Introduction
 Preliminaries
 Gaeta's theorem
 Divisors on an ACM subscheme of projective spaces
 Gorenstein ideals and Gorenstein liaison
 CIliaison invariants
 Geometric applications of the CIliaison invariants
 Glicci curves on arithmetically CohenMacaulay surfaces
 Unobstructedness and dimension of families of subschemes
 Dimension of families of determinantal subschemes
 Bibliography
