Hadamard Products of Projective Varieties (Frontiers in Mathematics)

Foreword
Preface
Acknowledgements
Contents
1 Hadamard Products
	1.1 Basic Facts
	1.2 Hadamard Transformations
2 Linear Spaces
	2.1 Multiplying by a Point
	2.2 Powers of a Line
		2.2.1 Equations
	2.3 Powers of Linear Spaces
	2.4 Hyperplanes
	2.5 The Tropical Point of View
		2.5.1 Preliminaries
		2.5.2 Dimension Formula
		2.5.3 Degree Formula
	2.6 Linear Spaces and Segre Embeddings
3 Not Generic Cases in P2
	3.1 Sets of Collinear Points in P2
	3.2 Points and Lines
4 Grids and Rulings
	4.1 Grids in P2
	4.2 Rulings
	4.3 Points and Lines in Pn
	4.4 Hadamard Fat Grids
5 Degenerate Varieties
	5.1 Large Ambient Space
	5.2 Small Ambient Space
	5.3 Some Examples
6 Hypersurfaces
	6.1 Hadamard Products of Hypersurfaces
	6.2 Hadamard Powers of Hypersurfaces
7 Binomial Varieties
	7.1 Two First Results
	7.2 A More Systematic Approach
		7.2.1 The Varieties of Hadamard Transformations
8 Hilbert Functions
	8.1 Sets of Points with the Same Cardinality in P3
	8.2 Points and Lines in P3
	8.3 Points and Lines in Pn
	8.4 Two Generic Linear Spaces
	8.5 Degenerate Varieties
	8.6 Binomial Varieties with the Same Binomial Exponents
9 Star Configurations
	9.1 Star Configurations of Points
	9.2 General Star Configurations
	9.3 Hadamard Star Configurations
		9.3.1 Hadamard Star Configurations: How to Find Them
		9.3.2 Hadamard Star Configurations in Pn: Can We Construct Them?
	9.4 Contact Star Configurations
		9.4.1 Contact Star Configurations and Hadamard Products
10 Gorenstein Sets of Points in P3
	10.1 Planar Complete Intersections
	10.2 Stick Figures of Lines
	10.3 Gorenstein Points in P3 from the h-vector
11 Pure Commutative Algebra
	11.1 Hadamard vs. Join
	11.2 Monomial Ideals
	11.3 Fat Points in Pn
	11.4 Graphs and Hypergraphs
		11.4.1 Toric Ideals of Graphs
12 Open Questions
	12.1 From X and Y to XY
	12.2 Identifiability and Secant Varieties
	12.3 Binomial Varieties
	12.4 Symbolic Powers
	12.5 Equations
	12.6 Hadamard Decompositions
A Background Material
	A.1 Polynomial Rings
	A.2 Ideals
	A.3 Monomial Orderings and Gröbner Bases
	A.4 Projective Geometry
	A.5 Morphisms
	A.6 Plücker Coordinates
	A.7 Segre and Veronese Embeddings
	A.8 Secant Varieties
	A.9 Hilbert Function, Castelnuovo–Mumford Regularity,and Betti Numbers
	A.10 Cohen–Macaulay and Gorenstein Ideals
	A.11 Liaison Theory
		A.11.1 Migliore and Nagel Construction
	A.12 Symbolic Powers of Ideals
	A.13 Fat Points in P1 x P1
B Our Procedures in Singular
	B.1 Computing Hadamard Products of Ideals
	B.2 Computing Hilbert Function of Ideals
	B.3 Computing Ideals of Points
	B.4 Creating Collinear Points
C The Package hadamard.m2
	C.1 Product of Points
	C.2 hadamardProduct
	C.3 hadamardPower
References
Index