Power Sums, Gorenstein Algebras, and Determinantal Varieties
by A. Iarrobino and V. Kanev
Appendix: The Gotzmann Theorems and the Hilbert Scheme,
by A. Iarrobino and S. L. Kleiman.
Springer Lecture Notes #1721, 346+xxix pages, December, 1999.
Table of Contents
Introduction, Informal History and Brief Outline
- Canonical forms and catalecticant matrices of higher partial derivatives of a form
- Apolarity and Artinian Gorenstein algebras
- Families of sets of points
- Brief summary of chapters
Part I. Catalecticant Varieties
Chapter 1. Forms and Catalecticant Matrices
- Apolarity and catalecticant varieties: the dimensions of the vector spaces of higher
partials.
- Determinantal loci of the first catalecticant, the Jacobian
- Binary forms and Hankel matrices
- Detailed summary and preparatory results
Chapter 2. Sums of Powers of Linear Forms, and Gorenstein Algebras
- Waring's problem for general forms
- Uniqueness of additive decomposition
- The Gorenstein algebra of a homogeneous polynomial
Chapter 3. Tangent Spaces to Catalecticant Schemes
- The tangent space to the deteminantal scheme $V_s(u,v;r)$ of the catalecticant matrix
- The tangent space to the scheme $Gor(T)$ parametrizing forms with fixed dimensions of the
partials
Chapter 4. The locus $PS(s,j;r)$ of Sums of Powers, and Determinantal Loci of Catalecticant
Matrices
- The case $r=3$.
- Sets of $s$ points in $P^{r-1}$ and Gorenstein ideals
- Gorenstein ideals whose lowest degree generators are a complete intersection
- The smoothness and dimension of the scheme $Gor(T)$ when $r=3$, a survey
Part II. Catalecticant Varieties and the Punctual Hilbert Scheme
Chapter 5. Forms and Zero-Dimensional schemes, I: Basic results
- Annihilating scheme in $P^{r-1}$ of a form
- Flat families of zero-dimensional schemes and limit ideals
- Existence theorems for annihilating schemes when $r=3$
- Power sum representations in three and more variables
- Betti strata of the punctual Hilbert scheme
- The length of a form, and the closure of the locus $PS(s,j;3)$ of power sums
- Codimension three Gorenstein schemes in $P^n$
Chapter 6. Forms and Zero-Dimensional schemes, II: Annihilating Schemes and Reducible
$Gor(T)$
- Uniqueness of the annihilating scheme: closure of $PS(s,j;r)$
- Varieties $Gor(T), T=T(j,r)$ with several components
- Other reducible varieties $Gor(T)$
- Locally Gorenstein annihilating schemes
Chapter 7. Connectedness and Components of the Determinantal Locus $PV_s(u,v;r)$
- Connectedness of $PV_s(u,v;r)$
- The irreducible components of $V_s(u,v;r)$
- Multisecant varieties of the Veronese variety
- Power sum representations in three and more variables
Chapter 8. Closures of the variety $Gor(T)$, and the Parameter Space $G(T)$ of
Graded Algebras
Chapter 9. Questions and Problems
Appendix A. Divided Power Rings and Polynomial Rings
Appendix B. Height Three Gorenstein Ideals
- Pfaffian formulas
- Resolutions of height 3 Gorenstein ideals and their squares
- Resolutions of annihilating ideals of power sums
- Maximum Betti numbers, given $T$
Appendix C. The Gotzmann Theorems and the Hilbert Scheme (by A. Iarrobino and S.L.
Kleiman)
- Order sequences and Macaulay's Theorem on Hilbert functions
- Macaulay and Gotzmann polynomials
- Gotzmann's Persistence Theorem and $m$-Regularity
- The Hilbert scheme $Hilb^P(\mathbb P^{r-1}$
- Gorenstein sequences having a constant subsequence of maximal growth, and $Hilb^P(\mathbb
P^{r-1}$
Appendix D. Examples of "Macaulay" scripts
Appendix E. Concordance with the 1996 version
References (an extended bibliography)
Index
Index of names
Index of Notation (includes short definitions)
Book homepage
Directions to view or download Introduction
Short Description of Appendix: The Gotzmann Theorems and the Hilbert scheme (with
S.L. Kleiman)
drawing by Dad, for frontispiece of book
A. Iarrobino web page
NU Math Dept web page
Last modified: January 27, 2000.
URL:http://www.math.neu.edu/~iarrobino/PScontents.html.