Ideals Of Powers And Powers Of Ideals: Intersecting Algebra, Geometry, And Combinatorics (lecture Notes Of The Unione Matematica Italiana)

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.--Provided by publisher Foreword Preface Acknowledgements Participants of PRAGMATIC 2017 Lecturers and Collaborators Organizers Students Contents Part I Associated Primes of Powers of Ideals 1 Associated Primes of Powers of Ideals 1.1 Associated Primes of Modules 1.2 Reducing the Problem 1.3 Sketch of the Missing Details 1.4 Final Comments 2 Associated Primes of Powers of Squarefree Monomial Ideals 2.1 General (Useful) Facts About Monomial Ideals 2.2 Monomial Ideals and Connections to Graph Theory: A First Look 2.3 The Index of Stability 2.4 Persistence of Primes 2.5 Elements of ass(Is) 3 Final Comments and Further Reading Part II Regularity of Powers of Ideals 4 Regularity of Powers of Ideals and the Combinatorial Framework 4.1 Regularity of Powers of Ideals: The General Question 4.2 Squarefree Monomial Ideals and Combinatorial Framework 4.2.1 Simplicial Complexes 4.2.2 Hypergraphs 4.2.3 Stanley-Reisner Ideals and Edge Ideals 4.3 Hochster's and Takayama's Formulas 5 Problems, Questions, and Inductive Techniques 5.1 Regularity of Powers of Edge Ideals 5.2 Regularity of Symbolic Powers of Edge Ideals 5.3 Inductive Techniques 6 Examples of the Inductive Techniques 6.1 Proof of Theorem 5.8 6.2 Proof of Theorem 5.1 6.3 Proof of Theorem 5.2 7 Final Comments and Further Reading Part III The Containment Problem 8 The Containment Problem: Background 8.1 Fat Points 8.2 Blow Ups and Sheaf Cohomology 8.3 Hilbert Functions 8.4 Waldschmidt Constants: Asymptotic α 9 The Containment Problem 9.1 Containment Problems 9.2 The Resurgence 10 The Waldschmidt Constant of Squarefree Monomial Ideals 10.1 The Waldschmidt Constant (General Case) 10.2 The Squarefree Monomial Case 10.3 Connection to Graph Theory 11 Symbolic Defect 11.1 Introducing the Symbolic Defect 11.2 Some Basic Properties 11.3 Computing sdefect(I,m) for Star Configurations 11.4 A Connection to the Containment Problem 12 Final Comments and Further Reading Part IV Unexpected Hypersurfaces 13 Unexpected Hypersurfaces 13.1 The SHGH Conjecture 13.2 A More General Problem 13.3 Unexpected Curves and BMSS Duality 13.4 Cones 14 Final Comments and Further Reading Part V Waring Problems 15 An Introduction to the Waring Problem 15.1 Waring Problems for Homogeneous Polynomials 15.2 Existence Questions 16 Algebra of the Waring Problem for Forms 16.1 Apolarity 16.2 The Apolarity Lemma 16.3 Waring Rank 16.4 A Sketch of a Proof of the Apolarity Lemma 17 More on the Waring Problem 17.1 Maximal Waring Rank 17.2 It is More Complex Over the Reals 17.3 Waring Loci 18 Final Comments and Further Reading Part VI PRAGMATIC Material 19 Proposed Research Problems 19.1 Project 1: The Waldschmidt Constant of Monomial Ideals 19.2 Project 2: The Symbolic Defect of Monomial Ideals 19.3 Project 3: Regularity of Powers of Ideals 19.4 Project 4: Beyond Perfect Graphs 19.5 Project 5: Resurgences for Fat Points 19.6 Project 6: Unexpected Curves 20 The Art of Research 20.1 Jump In! 20.2 How to Read a Paper When You Feel You Must 20.3 Do Experiments and Make Examples 20.4 Make Guesses 20.5 Write Proofs for Special Cases 20.6 Develop Parallel Questions 20.7 Writing Papers 20.8 Collaboration 20.9 Presenting Your Work References Index "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."-- prové de l'editor