Hartshorne's Algebraic Geometry: Varieties

Robin Hartshorne's "Algebraic Geometry" stands as one of the most influential textbooks in modern algebraic geometry. Published in 1977, this textbook revolutionized the teaching and learning of algebraic geometry by providing a systematic introduction to the subject. This page explores the fundamental concepts of varieties as presented in Hartshorne's work, providing insights into this rich mathematical field.

Introduction to Algebraic Varieties

Algebraic varieties represent geometric objects that can be described as the solution sets to systems of polynomial equations. They form the central objects of study in classical algebraic geometry, bridging the fields of algebra and geometry. Hartshorne's approach to varieties begins with affine varieties, then progresses to projective varieties, and eventually to abstract varieties and schemes.

The concept of algebraic varieties emerged from the study of curves and surfaces in the 19th century, but it was only in the 20th century that a suitable abstract framework was developed. Hartshorne's presentation builds upon the foundations laid by mathematicians like Zariski, Weil, and especially Grothendieck, who revolutionized the field through the introduction of schemes.

Affine Varieties

Affine varieties serve as the most basic type of algebraic varieties. They exist in affine space and are defined as the common zeros of a collection of polynomials.

For Hartshorne, the study of affine varieties begins with the Zariski topology, a topology on affine space defined using algebraic conditions rather than metric properties. In the Zariski topology, the closed sets are precisely the algebraic setssets of points where a collection of polynomials vanish.

Regular Functions and Coordinate Rings

An essential concept in Hartshorne's treatment is the notion of regular functions on varieties. These functions are locally given by quotients of polynomials where the denominator does not vanish.

The collection of all regular functions on a variety V forms a ring, denoted by O(V). This ring of regular functions plays a crucial role in understanding the geometric properties of V through the dictionary between geometry and algebra.

This fundamental result establishes a correspondence between geometric objects (varieties) and algebraic objects (quotients of polynomial rings). The coordinate ring contains important geometric information, such as the dimension of the variety and the singular points.

Morphisms Between Varieties

Hartshorne defines morphisms between varieties as functions that are locally given by regular functions. These morphisms preserve the geometric structure and allow for meaningful comparisons between different varieties.

Morphisms between varieties naturally induce homomorphisms between their rings of regular functions, creating a correspondence between geometric maps and algebraic homomorphisms that goes in the opposite direction (contravariant).

Projective Varieties

Projective varieties form another central theme in Hartshorne's work. Unlike affine varieties, which live in ordinary affine space, projective varieties exist in projective space, which can be thought of as affine space with a "hyperplane at infinity" added.

Projective varieties possess several desirable properties that affine varieties lack. For instance, any two lines in intersect (including at points at infinity), which is not the case in the affine plane. Projective varieties are also always "complete" in an appropriate sense, analogous to compactness in topology.

Dimension Theory

Dimension theory forms a crucial part of Hartshorne's treatment of varieties. The dimension of a variety captures its intuitive geometric dimension and is defined algebraically using chains of distinct irreducible subvarieties.

Several equivalent definitions of dimension exist, each connecting geometry to algebra. For instance, the dimension of an affine variety equals the Krull dimension of its coordinate ring, which is defined in terms of chains of prime ideals.

Understanding the dimension of varieties is fundamental to classification and to studying their properties. Hartshorne provides numerous examples where dimension theory reveals beautiful connections between geometric intuition and precise algebraic definitions.

Singularities

Another fundamental concept is that of singularities, which are points where a variety fails to look like standard affine space locally. The study of singularities occupies an important place in Hartshorne's treatment.

The set of singular points forms a proper closed subset of any variety. A variety without any singular points is called smooth or nonsingular. Many of Hartshorne's exercises and examples focus on identifying and analyzing singularities, which play a vital role in classification theories and provide insight into the geometry of varieties.

Abstract Varieties

Hartshorne eventually generalizes the concept of varieties beyond affine and projective spaces, leading to the notion of abstract (or pre) varieties. This abstraction allows for greater flexibility while retaining the geometric intuition.

This approach allows Hartshorne to construct more complex geometric objects by gluing together simpler affine pieces, much as one can construct surfaces by gluing patches. The abstract perspective removes the need to think of varieties as embedded in some ambient space, instead considering them as geometric objects in their own right.

Sheaves and Cohomology

One of Hartshorne's most significant contributions to textbook exposition is the introduction of sheaf theory and cohomology into elementary algebraic geometry. These tools provide powerful mechanisms for studying global properties of varieties.

The sheaf of regular functions O_X, for instance, assigns to each open set U the ring of regular functions on U. This structure allows mathematicians to track how functions behave locally and globally. Sheaf cohomology groups H^i(X, F) measure global obstructions to solving local problems.

The Riemann-Roch Theorem

A highlight of Hartshorne's treatment of varieties is his discussion of the Riemann-Roch theorem for curves, a fundamental result linking geometric and cohomological data.

This theorem has far-reaching consequences in the study of curves and provides insight into the interplay between algebraic and topological properties of varieties. Hartshorne's exposition makes clear how this classical result can be understood and generalized using modern techniques of sheaf cohomology.

Conclusion

Hartshorne's "Algebraic Geometry" revolutionized the teaching and learning of this field by providing a systematic development from classical varieties to modern schemes. His treatment of varieties serves as a foundation for understanding more advanced concepts in algebraic geometry, including schemes, moduli spaces, and birational geometry.

The study of varieties continues to be an active area of research with connections to many other branches of mathematics, including number theory, complex analysis, and mathematical physics. Hartshorne's exposition remains a valuable resource for both students and researchers, offering deep insights into the elegant and powerful world of algebraic varieties.