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E-raamat: Ideals of Powers and Powers of Ideals: Intersecting Algebra, Geometry, and Combinatorics

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Ideals of Powers and Powers of Ideals: Intersecting Algebra, Geometry, and CombinatoricsSuurem pilt
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This book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning  our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms.  Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes.

Arvustused

This is a very interesting monograph providing a fast introduction to different fields of research devoted to modern aspects and develompents of commutative algebra, algebraic geometry, combinatorics, etc. (Piotr Pokora, zbMATH 1445.13001, 2020)

Part I Associated Primes of Powers of Ideals
1 Associated Primes of Powers of Ideals
3(10)
1.1 Associated Primes of Modules
4(1)
1.2 Reducing the Problem
5(2)
1.3 Sketch of the Missing Details
7(4)
1.4 Final Comments
11(2)
2 Associated Primes of Powers of Squarefree Monomial Ideals
13(20)
2.1 General (Useful) Facts About Monomial Ideals
14(4)
2.2 Monomial Ideals and Connections to Graph Theory: A First Look
18(2)
2.3 The Index of Stability
20(1)
2.4 Persistence of Primes
21(5)
2.5 Elements of ass(Is)
26(7)
3 Final Comments and Further Reading
33(4)
Part II Regularity of Powers of Ideals
4 Regularity of Powers of Ideals and the Combinatorial Framework
37(14)
4.1 Regularity of Powers of Ideals: The General Question
37(3)
4.2 Squarefree Monomial Ideals and Combinatorial Framework
40(6)
4.2.1 Simplicial Complexes
41(1)
4.2.2 Hypergraphs
42(2)
4.2.3 Stanley-Reisner Ideals and Edge Ideals
44(2)
4.3 Hochster's and Takayama's Formulas
46(5)
5 Problems, Questions, and Inductive Techniques
51(8)
5.1 Regularity of Powers of Edge Ideals
51(3)
5.2 Regularity of Symbolic Powers of Edge Ideals
54(1)
5.3 Inductive Techniques
55(4)
6 Examples of the Inductive Techniques
59(8)
6.1 Proof of Theorem 5.8
59(2)
6.2 Proof of Theorem 5.1
61(3)
6.3 Proof of Theorem 5.2
64(3)
7 Final Comments and Further Reading
67(4)
Part III The Containment Problem
8 The Containment Problem: Background
71(6)
8.1 Fat Points
72(1)
8.2 Blow Ups and Sheaf Cohomology
72(1)
8.3 Hilbert Functions
72(1)
8.4 Waldschmidt Constants: Asymptotic α
73(4)
9 The Containment Problem
77(4)
9.1 Containment Problems
77(2)
9.2 The Resurgence
79(2)
10 The Waldschmidt Constant of Squarefree Monomial Ideals
81(8)
10.1 The Waldschmidt Constant (General Case)
81(1)
10.2 The Squarefree Monomial Case
82(4)
10.3 Connection to Graph Theory
86(3)
11 Symbolic Defect
89(10)
11.1 Introducing the Symbolic Defect
89(2)
11.2 Some Basic Properties
91(1)
11.3 Computing sdefect(I, m) for Star Configurations
92(5)
11.4 A Connection to the Containment Problem
97(2)
12 Final Comments and Further Reading
99(4)
Part IV Unexpected Hypersurfaces
13 Unexpected Hypersurfaces
103(8)
13.1 The SHGH Conjecture
104(1)
13.2 A More General Problem
105(3)
13.3 Unexpected Curves and BMSS Duality
108(2)
13.4 Cones
110(1)
14 Final Comments and Further Reading
111(4)
Part V Waring Problems
15 An Introduction to the Waring Problem
115(6)
15.1 Waring Problems for Homogeneous Polynomials
116(2)
15.2 Existence Questions
118(3)
16 Algebra of the Waring Problem for Forms
121(6)
16.1 Apolarity
121(2)
16.2 The Apolarity Lemma
123(1)
16.3 Waring Rank
124(1)
16.4 A Sketch of a Proof of the Apolarity Lemma
125(2)
17 More on the Waring Problem
127(6)
17.1 Maximal Waring Rank
127(1)
17.2 It is More Complex Over the Reals
128(1)
17.3 Waring Loci
129(4)
18 Final Comments and Further Reading
133(4)
Part VI PRAGMATIC Material
19 Proposed Research Problems
137(6)
19.1 Project 1: The Waldschmidt Constant of Monomial Ideals
137(1)
19.2 Project 2: The Symbolic Defect of Monomial Ideals
138(1)
19.3 Project 3: Regularity of Powers of Ideals
139(1)
19.4 Project 4: Beyond Perfect Graphs
140(1)
19.5 Project 5: Resurgences for Fat Points
141(1)
19.6 Project 6: Unexpected Curves
141(2)
20 The Art of Research
143(6)
20.1 Jump In!
143(1)
20.2 How to Read a Paper When You Feel You Must
144(1)
20.3 Do Experiments and Make Examples
144(1)
20.4 Make Guesses
145(1)
20.5 Write Proofs for Special Cases
145(1)
20.6 Develop Parallel Questions
146(1)
20.7 Writing Papers
146(1)
20.8 Collaboration
147(1)
20.9 Presenting Your Work
147(2)
References 149(8)
Index 157
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