Singular Loci of Schubert Varieties - Clear Guide to Singular Loci

In this review of Singular Loci of Schubert Varieties the reviewer finds a focused, research-level treatment aimed at readers who need a consolidated account of singular loci in Schubert varieties. The single biggest reason to buy is its restructured synthesis of two decades of research, which brings together methods and computations that are otherwise scattered across many journal articles. For graduate students and researchers needing concrete computations and several methodological viewpoints, this book provides clarity and worked examples rather than a broad textbook survey.

Key Features

• Consolidated research: Presents a restructured account of many research articles, making complex literature on singular loci more accessible to readers who need a single reference.
• Computation focus: Emphasizes explicit computations for the singular loci of Schubert varieties and their tangent spaces, supporting hands-on study and verification.
• Multiple methods: Uses standard monomial theory, the nil Hecke ring, and Kazhdan-Lusztig theory to offer varied approaches that illuminate different aspects of singular loci.
• New results included: Contains new results presented alongside classical material, helping readers see current developments in the subdiscipline.
• Worked examples: Includes sufficient examples to emphasize key points, which aids in understanding abstract constructions and concrete calculations.

Who It's For

This book is best for graduate students, postdocs, and researchers in representation theory, algebraic geometry, and combinatorics who require a concentrated study of singular loci in Schubert varieties and the computational tools to work with them. It suits readers who already have background in algebraic geometry or related areas and want a focused reference rather than an introductory text.

Those seeking a gentle introduction to algebraic geometry or a broad survey of Schubert calculus should look elsewhere; this work assumes familiarity with advanced techniques and concentrates on the specific topic of singular loci and tangent space computations.

Pros & Cons

Pros

• Carefully synthesizes two decades of literature into a single, coherent presentation that saves research time.
• Provides concrete computations and examples that make abstract theory more usable in practice.
• Combines several powerful methods, giving readers multiple tools to approach problems about singular loci.

Cons

• Not an introductory textbook; readers without background in algebraic geometry or representation theory may find it dense.

Specifications

Title	Singular Loci of Schubert Varieties
Series	Progress in Mathematics, 182
Authors	Sara Billey, V. Lakshmibai
Primary topics	Singular loci, Schubert varieties, tangent spaces
Methods used	Standard monomial theory, nil Hecke ring, Kazhdan-Lusztig theory
Content focus	Computations, examples, and new results

Our Verdict

Singular Loci of Schubert Varieties is a valuable, specialized reference for researchers and advanced students who need a consolidated, computation-oriented treatment of singular loci. Its combination of methods and worked examples makes it good value for those requiring depth on this subdiscipline, while beginners should supplement it with more introductory texts before diving in.

Frequently Asked Questions

Does this book include worked examples?
Yes, the book presents sufficient examples alongside new results to emphasize key points in computations of singular loci.

What mathematical background is required?
Readers should have prior exposure to algebraic geometry, representation theory, or combinatorics to fully benefit from the material.

Which methods are emphasized?
The text uses standard monomial theory, the nil Hecke ring, and Kazhdan-Lusztig theory as central tools for studying singular loci.