Determinantal Ideals of Square Linear Matrices

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.

“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)

• Part I: General oversight.- Background steps in determinantal rings.- Algebraic preliminaries.- Geometric oversight.- Part II: Linear section of notable structured square matrices.- Linear sections of the generic square matrix.- Symmetry preserving linear sections of the generic symmetric matrix.- Linear sections of the generic square Hankel matrix.- Hankel like catalecticants.- The dual variety of a linear determinantal hypersurface.- Part III: Other classes of linear sections.- Hilbert-Burch linear sections.- Apocryphal classes.- Appendix.- Index.