Algebraic Geometry by Robin Hartshorne

Introduction

Since its first publication in 1977, Robin Hartshornes Algebraic Geometry has become the standard graduatelevel textbook for the subject. Written in a clear, rigorous style, the book introduces the language of schemes and provides a bridge between classical algebraic geometry (based on varieties) and the modern, functorial approach of Grothendieck. This page offers a concise but thorough overview for students deciding whether to study Hartshorne, as well as for researchers seeking a quick reminder of the books organization.

About the Book

The full title is Algebraic Geometry, published by Springer in the Graduate Texts in Mathematics series (volume 52). It is often referred to simply as Hartshorne. The author, Robin Hartshorne, earned his Ph.D. under the supervision of Alexandre Grothendieck and later became a professor at the University of Illinois. His deep insight into the foundations of the subject is evident throughout the text, which combines concise definitions, careful proofs, and a large selection of exercises that range from routine calculations to challenging researchstyle problems.

Structure & Organization

The book is divided into three parts, each building on the previous one.

Part I Varieties

Chapter1 treats affine and projective varieties over an algebraically closed field. The material is familiar to anyone who has studied classical algebraic geometry, but Hartshorne reorganizes it to highlight the role of coordinate rings and sheaves. Important concepts such as dimension, regular functions, and morphisms are introduced in a way that prepares the reader for the more abstract theory of schemes.

PartII Schemes

Chapters2 and3 constitute the heart of the book. Chapter2 defines locally ringed spaces and explains how schemes arise by gluing spectra of rings. The author emphasizes the geometric intuition behind the Zariski topology and the role of the structure sheaf. Chapter3 develops the basic cohomology theory of coherent sheaves, introduces the notion of separated and proper morphisms, and culminates in Serres criterion for affineness. The results in these chapters are essential for anyone wishing to work in modern algebraic geometry.

PartIII Cohomology

Chapter4 is dedicated to sheaf cohomology on projective space, culminating in the celebrated theorem of Serre on the vanishing of higher cohomology for large twists. Chapter5 applies this machinery to prove the RiemannRoch theorem for curves, the theorem of RiemannRoch for surfaces (via the adjunction formula), and several deep results such as the HirzebruchRiemannRoch formula for smooth projective varieties. The exposition culminates with a discussion of intersection theory and several examples that demonstrate the power of the cohomological viewpoint.

Key Topics Covered

• Sheaf Theory Introduction of presheaves, sheaves, stalks, and the sheafification process.
• Spec and Proj Construction of affine and projective schemes from rings and graded rings.
• Morphisms of Schemes Separated, proper, finite, and tale morphisms with geometric interpretations.
• Coherent Sheaves Definition, basic properties, and the correspondence with finitely generated modules.
• ech Cohomology Computation of cohomology groups via open covers, leading to the development of sheaf cohomology.
• Serre Duality Statement and proof for projective varieties; applications to canonical bundles.
• RiemannRoch Theorems Both the classical curve case and the higherdimensional version.
• Intersection Theory Preliminary treatment of intersection numbers, divisors, and the Chow ring.
• Examples and Exercises A wide range of problems, including computations on curves, surfaces, and higherdimensional varieties.

Why the Book Matters

Hartshornes text is more than a collection of definitions; it offers a unified perspective that links algebraic geometry with commutative algebra, complex geometry, and number theory. Its influence can be seen in the way graduate curricula are organized worldwide and in the many research papers that cite it as a primary reference. The books emphasis on cohomology has also helped shape modern approaches to moduli problems, deformation theory, and birational geometry.

Critics sometimes point out that the exposition can be terse and that the exercises may be demanding for newcomers. Nevertheless, the clarity of the proofs and the depth of the material make it an indispensable resource for anyone serious about mastering algebraic geometry.

Tips for the First Reader

1. Refresh Commutative Algebra. A solid grasp of Noetherian rings, modules, and primary decomposition (as found in AtiyahMacdonald or Matsumura) will make Chapter2 much smoother.
2. Work Through the Exercises. The problems are carefully chosen to reinforce concepts; even the optional ones often contain the seeds of deeper results.
3. Use Supplementary Sources. Texts such as EisenbudHarris The Geometry of Schemes or GrtzWedhorn Algebraic Geometry I can provide alternative explanations for topics that feel opaque on a first reading.
4. Form a Study Group. Discussing proofs and examples with peers helps illuminate subtle points, especially in the cohomology chapters.
5. Dont Skip the Classical Parts. The early chapters on varieties give valuable geometric intuition that underpins the abstract language of later sections.

Further Resources

For readers who wish to extend their study beyond Hartshorne, the following resources are recommended:

• Hartshorne Residues and Duality A companion volume focusing on duality theory.
• G. Kempf, et al. Algebraic Geometry I A more gentle introduction to schemes.
• R. Vakil The Rising Sea Free online notes that expand on Hartshornes material with many additional examples.
• J. Kollr Rational Curves on Algebraic Varieties Shows how modern techniques build on the foundation laid by Hartshorne.

Conclusion

Robin Hartshornes Algebraic Geometry remains a cornerstone of graduate mathematics. Its rigorous treatment of schemes, coherent sheaves, and cohomology provides a solid foundation for both theoretical research and advanced coursework. While the book demands commitment and a strong background in commutative algebra, the intellectual rewards are substantial. Whether you are beginning your journey in algebraic geometry or seeking a reliable reference for deeper investigations, Hartshornes text is an essential companion.