Ideals of Powers and Powers of Ideals
Intersecting Algebra, Geometry, and Combinatorics
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Beschreibung
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.
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.
First book to contain a summary of known results on the associated primes of edge ideals First book to contain a summary of known results on the regularity of powers of monomial ideals Contains open problems for graduate students With a chapter for early career researchers on "how to do mathematics research" Provides an up-to-date and comprehensive list of references of papers in the area Written by authors who have made numerous contributions to these areas First book to introduce the ideal containment problem (e.g. Waldschmidt contstant resurgence symbolic defect)
Autor*in
Enrico Carlini
Themen in »Ideals of Powers and Powers of Ideals«
Associated Primes Castelnuovo-Mumford Regularity Containment Problem Edge and Cover Ideals of Finite Simple Graphs Monomial Ideals Powers of Ideals Symbolic Defect Symbolic Powers of Ideals Unexpected Curves and Hypersurfaces Waldschmidt Constant Waring Problems
Stimmen zu »Ideals of Powers and Powers of Ideals«
“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)
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