Determinantal Ideals of Square Linear Matrices

Zaqueu Ramos, Aron Simis

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English

Springer International Publishing AG
04 June 2024

Algebra; Algebraic geometry

This book explores determinantal ideals of square matrices from the perspective of commutative algebra, with a particular emphasis on linear matrices. Its content has been extensively tested in several lectures given on various occasions, typically to audiences composed of commutative algebraists, algebraic geometers, and singularity theorists. Traditionally, texts on this topic showcase determinantal rings as the main actors, emphasizing their properties as algebras. This book follows a different path, exploring the role of the ideal theory of minors in various situations—highlighting the use of Fitting ideals, for example. Topics include an introduction to the subject, explaining matrices and their ideals of minors, as well as classical and recent bounds for codimension. This is followed by examples of algebraic varieties defined by such ideals. The book also explores properties of matrices that impact their ideals of minors, such as the 1-generic property, explicitly presenting a criterion by Eisenbud. Additionally, the authors address the problem of the degeneration of generic matrices and their ideals of minors, along with applications to the dual varieties of some of the ideals. Primarily intended for graduate students and scholars in the areas of commutative algebra, algebraic geometry, and singularity theory, the book can also be used in advanced seminars and as a source of aid. It is suitable for beginner graduate students who have completed a first course in commutative algebra.

By: Zaqueu Ramos, Aron Simis
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 2024 ed.
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9783031552830
ISBN 10: 3031552830
Pages: 318
Publication Date: 04 June 2024
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Zaqueu Ramos is a Professor at the Federal University of Sergipe, Brazil. He holds a bachelor's degree in Mathematics from the Federal University of Sergipe, Brazil and a PhD degree in Mathematics from the Federal University of Pernambuco (2012). He completed his postdoctorate studies at the Federal University of Paraíba (2014-2015) under the supervision of Aron Simis. His research focuses on commutative algebra and its interactions with algebraic geometry. Aron Simis is an Emeritus Full Professor at the Federal University of Pernambuco, Brazil. He earned his PhD from Queen's University, Canada, under the supervision of Paulo Ribenboim. He previously held a full professorship at IMPA, Rio de Janeiro, Brazil. He was President of the Brazilian Mathematical Society (1985-1987) and a member, on several occasions, of international commissions of the IMU (International Mathematical Union) and TWAS (Academy of Sciences for the Developing World). His main research interests include main structures in commutative algebra; projective varieties in algebraic geometry; aspects of algebraic combinatorics; special graded algebras; foundations of Rees algebras; cremona and birational maps; algebraic vector fields; and differential methods.

Reviews for Determinantal Ideals of Square Linear Matrices

“This book delves into the algebraic and geometric properties of determinantal ideals of matrices. The matrices considered are typically derived from a generic matrix of variables by replacing some entries with linear forms. These matrices belong to a class referred to as linear sections in the text. The book is organised into three parts with ten chapters and an appendix. Each chapter begins with an abstract and concludes with a list of exercises and bibliography."" (Jorge Neves. zbMATH 1553.13001, 2025)