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Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness
Jan O. Kleppe, Oslo University College, Norway, Juan C. Migliore, University of Notre Dame, IN, Rosa Miró-Roig, University of Barcelona, Spain, Uwe Nagel, University of Paderborn, Germany, and Chris Peterson, Colorado State University, Fort Collins, CO
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Memoirs of the American Mathematical Society
2001; 116 pp; softcover
Volume: 154
ISBN-10: 0-8218-2738-3
ISBN-13: 978-0-8218-2738-3
List Price: US$49
Individual Members: US$29.40
Institutional Members: US$39.20
Order Code: MEMO/154/732
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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 \(t\times (t+c-1)\) homogeneous matrix), is in the Gorenstein liaison class of a complete intersection. Then Gorenstein liaison (G-liaison) theory is developed as a theory of generalized divisors on arithmetically Cohen-Macaulay schemes. In particular, a rather general construction of basic double G-linkage is introduced, which preserves the even G-liaison 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 Cohen-Macaulay 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
  • CI-liaison invariants
  • Geometric applications of the CI-liaison invariants
  • Glicci curves on arithmetically Cohen-Macaulay surfaces
  • Unobstructedness and dimension of families of subschemes
  • Dimension of families of determinantal subschemes
  • Bibliography
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