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Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness
 
Jan O. Kleppe Oslo University College, Oslo, Norway
Juan C. Migliore University of Notre Dame, Notre Dame, IN
Rosa Miró-Roig University of Barcelona, Barcelona, Spain
Uwe Nagel University of Paderborn, Paderborn, Germany
Chris Peterson Colorado State University, Fort Collins, CO
Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness
eBook ISBN:  978-1-4704-0325-6
Product Code:  MEMO/154/732.E
List Price: $52.00
MAA Member Price: $46.80
AMS Member Price: $31.20
Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness
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Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness
Jan O. Kleppe Oslo University College, Oslo, Norway
Juan C. Migliore University of Notre Dame, Notre Dame, IN
Rosa Miró-Roig University of Barcelona, Barcelona, Spain
Uwe Nagel University of Paderborn, Paderborn, Germany
Chris Peterson Colorado State University, Fort Collins, CO
eBook ISBN:  978-1-4704-0325-6
Product Code:  MEMO/154/732.E
List Price: $52.00
MAA Member Price: $46.80
AMS Member Price: $31.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1542001; 116 pp
    MSC: Primary 14;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. Preliminaries
    • 3. Gaeta’s theorem
    • 4. Divisors on an ACM subscheme of projective space
    • 5. Gorenstein ideals and Gorenstein liaison
    • 6. CI-liaison invariants
    • 7. Geometric applications of the CI-liaison invariants
    • 8. Glicci curves on arithmetically Cohen-Macaulay surfaces
    • 9. Unobstructedness and dimension of families of subschemes
    • 10. Dimension of families of determinantal subschemes
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1542001; 116 pp
MSC: Primary 14;

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.

  • Chapters
  • 1. Introduction
  • 2. Preliminaries
  • 3. Gaeta’s theorem
  • 4. Divisors on an ACM subscheme of projective space
  • 5. Gorenstein ideals and Gorenstein liaison
  • 6. CI-liaison invariants
  • 7. Geometric applications of the CI-liaison invariants
  • 8. Glicci curves on arithmetically Cohen-Macaulay surfaces
  • 9. Unobstructedness and dimension of families of subschemes
  • 10. Dimension of families of determinantal subschemes
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.