Determinantal Ideals of Square Linear Matrices - Zaqueu Ramos, Aron Simis

Determinantal Ideals of Square Linear Matrices

Buch | Hardcover
XX, 318 Seiten
2024 | 2024
Springer International Publishing (Verlag)
978-3-031-55283-0 (ISBN)
90,94 inkl. MwSt
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.

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. 

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.

Erscheinungsdatum
Zusatzinfo XX, 318 p. 5 illus.
Verlagsort Cham
Sprache englisch
Maße 155 x 235 mm
Themenwelt Mathematik / Informatik Mathematik Algebra
Mathematik / Informatik Mathematik Geometrie / Topologie
Schlagworte 1-generic matrix • Cohen-Macaulay • Determinantal ideals • dual variety • gradient ideal • Hessian determinant • homaloidal polynomial • ladder ideal • polar map • square linear matrices • submaximal minors
ISBN-10 3-031-55283-0 / 3031552830
ISBN-13 978-3-031-55283-0 / 9783031552830
Zustand Neuware
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