# Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness

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*Jan O. Kleppe; Juan C. Migliore; Rosa Miró-Roig; Uwe Nagel; Chris Peterson*

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.

#### Table of Contents

# Table of Contents

## Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness

- Contents vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. Preliminaries 918 free
- Chapter 3. Gaeta's Theorem 1221
- Chapter 4. Divisors on an ACM Subscheme of Projective Space 1928
- Chapter 5. Gorenstein Ideals and Gorenstein Liaison 2635
- Chapter 6. CI-Liaison Invariants 3544
- Chapter 7. Geometric Applications of the CI-Liaison Invariants 5261
- Chapter 8. Glicci curves on Arithmetically Cohen-Macaulay surfaces 6776
- Chapter 9. Unobstructedness and dimension of families of subschemes 7988
- Chapter 10. Dimension of families of determinantal subschemes 95104
- Bibliography 114123