Determinantal Ideals of Square Linear Matrices / by Zaqueu Ramos, Aron Simis.

Format:

Book

Author/Creator:

Ramos, Zaqueu.

Contributor:

Simis, Aron.

Language:

English

Subjects (All):

Commutative algebra.

Commutative rings.

Geometry, Algebraic.

Algebraic fields.

Polynomials.

Commutative Rings and Algebras.

Algebraic Geometry.

Field Theory and Polynomials.

Local Subjects:

Commutative Rings and Algebras.

Algebraic Geometry.

Field Theory and Polynomials.

Physical Description:

1 online resource (326 pages)

Edition:

1st ed. 2024.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2024.

Summary:

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.

Contents:

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.

ISBN:

3-031-55284-9