Algebraic Geometry and Arithmetic Curves: An Advanced Introduction to Schemes and Arithmetic Surfaces by Qing Liu

Introduction

Algebraic geometry stands as one of the most profound and elegant branches of modern mathematics, bridging abstract algebra with geometric intuition. For Indian students and researchers pursuing advanced studies in pure mathematics, understanding the theory of schemes and arithmetic curves is essential. Algebraic Geometry and Arithmetic Curves by Qing Liu offers a rigorous yet accessible gateway into this sophisticated domain. Published by OUP Oxford, this hardcover volume is a definitive resource for anyone looking to master the foundations of scheme theory and its applications to arithmetic surfaces and curve reduction.

Book Overview

This comprehensive text is divided into two major parts. The first part systematically introduces the theory of schemes, covering morphisms, base change, local properties such as normality and regularity, and culminates with Zariski's Main Theorem. It then moves into global aspects: coherent sheaves, cohomology groups, sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part concludes with the Riemann-Roch theorem and its application to smooth projective curves over a field, along with a detailed study of the Picard group for singular curves. The second part focuses on blowing-ups, desingularization of fibered surfaces over a Dedekind ring, intersection theory on arithmetic surfaces, Castelnuovo's criterion, and the existence of the minimal regular model. This structure makes the book ideal for both self-study and classroom use.

Key Highlights

• Rigorous Foundation: Builds from basic scheme theory to advanced topics like Grothendieck duality and arithmetic surfaces.
• Dual Focus: Balances abstract theory with concrete applications to arithmetic geometry and curve reduction.
• Complete Coverage: Includes singular curves, Picard groups, and desingularization techniques essential for research.
• Authoritative Publisher: Published by OUP Oxford, a hallmark of quality in academic mathematics.
• Durable Hardback: A sturdy volume designed for years of intensive use in libraries and personal collections.

Inside the Book

The book opens with a clear exposition of schemes, morphisms, and base change, ensuring readers with a solid background in commutative algebra can follow easily. Subsequent chapters delve into local properties, coherent sheaves, and cohomology, with careful proofs and illustrative examples. The treatment of sheaves of differentials and dualizing sheaves is particularly detailed, preparing the ground for Grothendieck's duality. The Riemann-Roch chapter is a highlight, showing how abstract theory yields powerful results for curves. In the second part, the author guides readers through blowing-ups and desingularization, using fibered surfaces over Dedekind rings as a central theme. Intersection theory on arithmetic surfaces is developed with precision, culminating in Castelnuovo's criterion and the construction of minimal regular models. Each chapter ends with exercises that range from routine to challenging, making it suitable for graduate courses.

Key Topics

• Schemes, morphisms, and base change
• Local properties: normality, regularity, Zariski's Main Theorem
• Coherent sheaves and cohomology groups
• Sheaves of differentials and dualizing sheaves
• Grothendieck's duality theory
• Riemann-Roch theorem for smooth projective curves
• Picard group and singular curves
• Blowing-ups and desingularization of fibered surfaces
• Intersection theory on arithmetic surfaces
• Castelnuovo's criterion and minimal regular models

Reader Benefits

By studying this book, readers will develop a deep, intuitive understanding of scheme theory and its arithmetic applications. The clear exposition helps bridge the gap between abstract algebra and geometric visualization. Indian students preparing for competitive exams or research in algebraic geometry will find the logical flow and comprehensive exercises invaluable. The book also serves as a reliable reference for working mathematicians, offering a self-contained treatment of both foundational and advanced topics. The focus on arithmetic curves and surfaces is particularly relevant for those interested in number theory and arithmetic geometry.

Learning Outcomes

• Master the language and tools of scheme theory, from definitions to advanced theorems.
• Understand the interplay between algebraic geometry and arithmetic through concrete examples.
• Apply cohomological methods to study curves and surfaces.
• Analyze singularities and desingularization techniques for fibered surfaces.
• Prove and use Castelnuovo's criterion and construct minimal regular models.

Who Should Read

This book is ideal for graduate students in mathematics who have completed a first course in commutative algebra and basic algebraic geometry. Researchers in algebraic geometry, arithmetic geometry, and number theory will find it an essential reference. Advanced undergraduates with a strong foundation in abstract algebra can also benefit from the early chapters. Indian students aiming for doctoral work in pure mathematics, especially those interested in the Mumbai school of algebraic geometry or the Tata Institute of Fundamental Research, will find this book aligns perfectly with their curriculum.

About the Author

Qing Liu is a renowned mathematician and professor at the Université de Bordeaux, France. His research focuses on arithmetic geometry, particularly the theory of curves, surfaces, and reduction. With a gift for clear exposition, Liu has trained generations of mathematicians. Algebraic Geometry and Arithmetic Curves reflects his deep expertise and pedagogical skill, making complex ideas accessible without sacrificing rigor. This book has become a standard text in graduate programs worldwide.

About the Publisher

OUP Oxford (Oxford University Press) is one of the oldest and most respected academic publishers in the world. Known for its rigorous editorial standards and commitment to scholarly excellence, OUP produces mathematics texts that are trusted by universities and research institutions globally. This hardcover edition maintains the high production quality expected from OUP, with clear typesetting, durable binding, and acid-free paper suitable for long-term use.

Conclusion

Algebraic Geometry and Arithmetic Curves is an indispensable resource for anyone serious about modern algebraic geometry. Whether you are a graduate student embarking on research or a seasoned mathematician seeking a comprehensive reference, Qing Liu's masterful treatment will deepen your understanding and inspire further exploration. Order your hardcover copy from Bookshops.in today and add this classic to your library.