1 Foundations
Algebraic geometry studies sets of solutions to polynomial equations and the spaces these solutions form. Its central idea is that geometric questions can be translated into algebraic ones, especially questions about rings, ideals, and polynomial relations. This perspective makes it possible to describe curves, surfaces, and more general spaces in a unified language.
Historically, the subject grew from classical studies of plane curves and surfaces. Modern algebraic geometry incorporates methods from commutative algebra, topology, and category theory, allowing mathematicians to treat objects of varying complexity with a common framework.
1.1 Polynomial equations and solution sets
A polynomial equation is an expression built from variables and coefficients using addition, multiplication, and nonnegative integer exponents. The set of points that satisfy one or more such equations is called a solution set. These sets may be finite, discrete, or continuous, and their shape depends on the degree and interaction of the equations.
In algebraic geometry, the same equation may be viewed over different number systems, such as the real numbers or complex numbers, producing different geometric behavior. This flexibility is one reason the subject is useful in both pure and applied mathematics.
1.2 Affine and projective varieties
Affine varieties arise as common zero sets of polynomial equations in ordinary affine space. They are the most direct geometric objects of classical algebraic geometry and are often used as local models for more complicated spaces.
Projective varieties are defined in projective space, where points at infinity are included to improve geometric behavior. This setting is especially useful for studying curves and surfaces without boundary effects and for obtaining compactness-like properties in a purely algebraic context.
1.3 Zariski topology
The Zariski topology is a topology adapted to algebraic sets. Its closed sets are defined by polynomial equations, so it is much coarser than the usual topology on Euclidean space. As a result, open sets tend to be large, and geometric reasoning often emphasizes algebraic containment rather than distance.
This topology is well suited to algebraic geometry because it reflects how varieties are built from polynomial constraints. It also provides a natural framework for defining irreducibility, generic points, and continuity of algebraic maps.
1.4 Coordinate rings
A coordinate ring records the polynomial functions on an affine algebraic set. It is formed by taking the polynomial ring and quotienting by the ideal of relations that vanish on the set. In this way, geometric information is encoded algebraically.
Many geometric properties can be read from the coordinate ring. For example, decomposition into factors corresponds to reducibility, while algebraic conditions on the ring often indicate smoothness, dimension, or singular behavior.
2 Basic objects
The basic objects of algebraic geometry are algebraic sets, varieties, and the maps between them. These objects capture the geometry of polynomial equations while retaining enough algebraic structure to allow precise calculations. They serve as the starting point for more advanced constructions such as schemes and moduli spaces.
2.1 Algebraic sets
An algebraic set is the common zero locus of a collection of polynomials. Such a set may consist of one or many components and can be described intrinsically by the equations that define it. Algebraic sets form the natural class of closed sets in affine algebraic geometry.
They may be studied through their defining ideals, which determine how many equations are needed and how the constraints interact. This correspondence is one of the central bridges between geometry and algebra.
2.2 Varieties
A variety is an algebraic set with additional conditions that make it behave more regularly, especially in classical treatments. Varieties are typically assumed to be reduced and often irreducible, depending on the convention being used. They provide the primary objects of study in many geometric problems.
Varieties can be examined locally or globally, and their structure often changes when viewed over different fields. The notion of variety captures the idea of a geometric space defined by polynomial data.
2.2.1 Irreducible varieties
An irreducible variety cannot be expressed as the union of two proper closed subvarieties. This property makes it a geometric analogue of an indecomposable algebraic object. Irreducibility is closely related to the primality of the defining ideal.
Irreducible varieties are important because many geometric arguments reduce to the irreducible case. They also provide the natural setting for generic behavior, where properties hold on a dense open subset.
2.2.2 Smooth varieties
A smooth variety has no singular points, meaning that it locally resembles ordinary affine space. Smoothness is often checked using derivatives or rank conditions on the Jacobian matrix. It is a key condition because many theorems are simplest in the smooth case.
Smooth varieties support well-behaved tangent spaces, differential forms, and local coordinate descriptions. They form the algebraic counterpart of manifolds in differential geometry, though the algebraic setting has its own distinctive features.
2.3 Morphisms of varieties
A morphism of varieties is a map defined by polynomial functions. Such maps preserve the algebraic structure of the objects involved and play the role of continuous maps in topology or smooth maps in differential geometry.
Morphisms allow one to compare varieties, transport functions between them, and study geometric constructions such as projections, embeddings, and finite coverings. They are central to the categorical viewpoint of the subject.
3 Commutative algebra background
Commutative algebra supplies the language and tools needed to study algebraic geometry. Since varieties and schemes are built from rings and ideals, many geometric questions become questions about algebraic structure. The interaction between algebra and geometry is especially strong in local analysis.
3.1 Rings and ideals
A ring is an algebraic system with addition and multiplication. Ideals are special subsets closed under addition and multiplication by arbitrary ring elements. They encode systems of equations and the relations among polynomial functions.
In algebraic geometry, ideals are used to define algebraic sets and to organize local and global structure. The passage from geometry to ideals and back is one of the subject’s foundational correspondences.
3.2 Prime and maximal ideals
A prime ideal is an ideal with a strong factorization property, while a maximal ideal is one that is not properly contained in any larger proper ideal. Prime ideals correspond to irreducible geometric features, and maximal ideals often correspond to points in affine space over an algebraically closed field.
These ideals help classify local and global behavior. They also form the basis for the construction of spectra, which are the building blocks of schemes.
3.3 Noetherian rings
A Noetherian ring is one in which every ascending chain of ideals stabilizes. This finiteness condition prevents infinitely increasing complexity in ideal structure and ensures that many arguments terminate.
Noetherian hypotheses are common in algebraic geometry because they guarantee manageable decompositions and support induction on dimension. They also underlie many important finiteness results.
3.4 Localization
Localization is a process that focuses on behavior near a chosen prime ideal or multiplicative set. It creates a ring in which certain elements become invertible, allowing local study of algebraic objects.
This technique is essential for analyzing varieties and schemes point by point. It makes it possible to isolate singularities, define local rings, and compare global information with local data.
3.5 Integral dependence and normalization
An element is integral over a ring if it satisfies a monic polynomial equation with coefficients in that ring. Integral dependence measures how one algebraic object extends another. Normalization is a process that replaces a ring or variety by a more regular version that is integrally closed in its field of fractions.
Normalization often simplifies singular or nonnormal behavior while preserving much of the original geometry. It is a standard tool in the study of singularities and birational geometry.
4 Schemes
Schemes generalize varieties by allowing more flexible local algebraic structure. They unify classical algebraic geometry with number theory and permit the systematic study of objects defined over arbitrary rings. In this framework, geometric spaces are assembled from local affine pieces and glued together.
4.1 Motivation for schemes
Schemes were introduced to extend geometric methods beyond varieties over algebraically closed fields. They allow one to work with arithmetic objects, such as integer rings, and to keep track of subtle local phenomena that classical varieties cannot capture.
This generalization provides a more powerful and versatile foundation. It also clarifies how geometry behaves under base change and how local algebra influences global structure.
4.2 Affine schemes
An affine scheme is the spectrum of a commutative ring. Its points are prime ideals, and its topology and structure are determined by algebraic data from that ring. This construction turns rings into geometric spaces.
Affine schemes are the basic building blocks of scheme theory. Many global arguments begin by reducing to the affine case, where calculations are more concrete.
4.3 Structure sheaf
The structure sheaf assigns to each open set a ring of functions defined on that set. It records local algebraic information and makes it possible to recover the ring-theoretic behavior of the scheme from its geometry.
The structure sheaf is crucial because the underlying topological space alone does not contain enough information. Together, the space and sheaf form the complete scheme.
4.4 Gluing affine schemes
Schemes are formed by gluing affine schemes along open subsets that match compatibly. This process is similar to assembling a manifold from coordinate charts, but the transition data are algebraic rather than smooth.
Gluing allows the construction of projective space, curves, and more complicated spaces from simple pieces. It reflects the local-to-global principle that is central to modern geometry.
4.5 Morphisms of schemes
A morphism of schemes is a map that respects both the topological space and the structure sheaf. Such morphisms generalize maps of varieties and encode algebraic behavior at every level of locality.
They are the natural arrows in scheme theory and are used to define fiber products, base change, and many other constructions. Morphisms also organize the relationship between geometry over different rings.
4.6 Closed and open subschemes
Closed subschemes are defined by sheaves of ideals and represent geometric subobjects with multiplicity or embedded structure. Open subschemes arise by restricting to an open subset and preserve the ambient scheme’s local structure.
These subschemes are fundamental because many geometric arguments reduce to understanding how spaces decompose into open and closed pieces. They also provide the language for local analysis and stratification.
5 Sheaves and cohomology
Sheaves and cohomology provide the tools for organizing local data and measuring global obstructions. They are indispensable in modern algebraic geometry because many important properties cannot be detected by local algebra alone. These methods link geometry, topology, and homological algebra.
5.1 Sheaves on topological spaces
A sheaf assigns compatible local data to open sets and describes how local information can be glued into global information. It formalizes the idea that functions, sections, or other algebraic objects may be studied locally and then assembled.
Sheaves capture both locality and compatibility. They are used not only in algebraic geometry but also in topology, analysis, and related fields.
5.2 Sheaves on varieties and schemes
On varieties and schemes, sheaves organize regular functions, modules, differential forms, and many other objects. They provide a precise way to track algebraic structures across different regions of a space.
This sheaf-theoretic viewpoint is especially powerful on schemes, where the structure sheaf and its modules encode much of the geometry. Many fundamental constructions are best expressed in sheaf language.
5.3 Cohomology of sheaves
Sheaf cohomology measures the extent to which local sections fail to extend globally. It detects hidden relationships among local pieces and often reveals obstructions to solving geometric problems.
Cohomology groups are used to study divisors, line bundles, deformations, and vanishing theorems. They are among the most important invariants in modern algebraic geometry.
5.4 Čech cohomology
Čech cohomology is a computational approach to cohomology based on an open cover of a space. It breaks a geometric object into overlapping regions and studies how local data match on intersections.
While not always sufficient as a complete theory, Čech cohomology is often useful for explicit calculations and for building intuition. It is especially helpful in situations where covers are manageable.
5.5 Derived functors in geometry
Derived functors extend constructions such as global sections to higher-dimensional cohomological information. They organize the failure of exactness in algebraic processes and are central to homological methods.
In geometry, derived functors explain many advanced results in a unified way. They connect sheaf cohomology with deeper tools from category theory and homological algebra.
6 Divisors and line bundles
Divisors and line bundles describe codimension-one geometry and the ways functions can twist globally. They are crucial in the study of curves, surfaces, and higher-dimensional varieties. These concepts also provide a bridge between local equations and global geometry.
6.1 Cartier divisors
A Cartier divisor is given locally by a single rational function, up to multiplication by units. It encodes hypersurface-like data and is well suited to smooth or mildly singular spaces.
Cartier divisors are closely tied to line bundles and often represent geometric conditions such as zeros and poles of functions. They are especially useful when divisor theory interacts with cohomology.
6.2 Weil divisors
A Weil divisor is a formal integer combination of codimension-one subvarieties. This definition is more combinatorial than that of Cartier divisors and works well on normal varieties.
Weil divisors are important in birational geometry and in the study of singular spaces. They help track how subvarieties intersect and how functions behave near them.
6.3 Line bundles and invertible sheaves
A line bundle is a family of one-dimensional vector spaces varying smoothly or algebraically over a variety or scheme. In algebraic geometry, line bundles correspond to invertible sheaves, which are sheaves locally isomorphic to the structure sheaf.
Line bundles measure twisting and play a central role in embedding varieties into projective space. They also appear in the formulation of divisors, cohomology, and moduli problems.
6.4 Picard group
The Picard group classifies line bundles up to isomorphism, with group law given by tensor product. It is an invariant that captures how line bundles combine and how they differ globally.
This group often reflects subtle geometry. For instance, it can detect whether certain divisors are principal or whether a space has nontrivial twisting.
6.5 Divisor class group
The divisor class group measures Weil divisors modulo linear equivalence. It generalizes the idea of classifying codimension-one data while accounting for principal divisors arising from rational functions.
It is particularly useful for singular varieties, where Cartier and Weil divisors may not coincide. The class group offers a way to organize divisor theory in a broader setting.
7 Curves
Algebraic curves are among the oldest and most thoroughly studied objects in algebraic geometry. Despite their one-dimensional nature, they exhibit rich behavior involving singularities, function fields, divisors, and moduli. Many major ideas in the subject first appear in the study of curves.
7.1 Algebraic curves
An algebraic curve is a one-dimensional variety or scheme, often defined by polynomial equations in one or more variables. Curves may be affine or projective, smooth or singular, and their global properties depend strongly on these distinctions.
Curves serve as a testing ground for broader theories. Their classification and invariants are often more accessible than those of higher-dimensional varieties.
7.2 Singularities of curves
A singularity is a point where a curve fails to be smooth. Common examples include nodes, cusps, and multiple branches meeting at a point. Singularities complicate local geometry but often encode important global information.
Studying singular points involves examining local equations and their tangent behavior. Resolving or classifying singularities is a major theme in the subject.
7.3 Riemann–Roch theorem
The Riemann–Roch theorem relates the dimension of spaces of global sections to the degree of a divisor and the geometry of the curve. It is one of the foundational results of algebraic geometry and complex curve theory.
This theorem provides a powerful tool for counting functions with prescribed zeros and poles. It also connects geometry with linear algebra through the language of cohomology.
7.4 Genus
The genus of a curve is a fundamental invariant that measures its complexity. For smooth projective curves, it can be interpreted in several equivalent ways, including topological, analytical, and algebraic definitions.
Curves of genus zero, one, and greater than one exhibit very different behavior. The genus influences the number of global functions, the form of divisors, and the structure of the curve’s automorphisms.
7.5 Elliptic curves
An elliptic curve is a smooth projective curve of genus one with a specified point. It has a group law defined geometrically, making it both a curve and an algebraic group.
Elliptic curves are central objects in number theory and algebraic geometry. They illustrate deep interactions among geometry, arithmetic, and the theory of rational points.
8 Surfaces and higher-dimensional varieties
Higher-dimensional varieties extend the ideas of curves into more complex settings. Surfaces already display phenomena absent from one-dimensional geometry, and dimensions three and above introduce further challenges. Classification and birational methods become especially significant in these cases.
8.1 Algebraic surfaces
An algebraic surface is a two-dimensional variety or scheme. Surfaces can have rich configurations of curves, singularities, and divisors, making them more intricate than curves but still tractable in many cases.
The study of surfaces includes their geometry, intersection patterns, and birational models. Many foundational results in classification theory were first developed for surfaces.
8.2 Birational geometry
Birational geometry studies varieties that become isomorphic after removing lower-dimensional subsets. It focuses on the field of rational functions and on transformations that preserve most of the geometric structure.
This viewpoint is especially useful for classifying varieties up to rational equivalence. It helps identify when different models represent the same underlying function field.
8.3 Resolution of singularities
Resolution of singularities is a process that replaces a singular variety by a related smooth or less singular one. The replacement is constructed through controlled geometric operations such as blowups.
This technique is valuable because it allows difficult spaces to be studied via better-behaved models. It also supports many proofs by reducing questions to the smooth case.
8.4 Minimal models
A minimal model is a birational representative that cannot be simplified further by certain contraction operations. The idea is to choose, within a birational class, a model that is in some sense economically structured.
Minimal model theory is one of the central tools in the classification of higher-dimensional varieties. It seeks canonical forms while preserving essential birational information.
8.5 Classification problems
Classification problems ask how varieties can be organized into families according to geometric invariants. For curves and surfaces, classification is often guided by genus, Kodaira dimension, singularities, and birational type.
In higher dimensions, classification becomes much more complicated, and broad structural principles replace complete lists. Even so, classification remains a driving force in the field.
9 Intersection theory
Intersection theory studies how subvarieties meet inside a larger ambient space. It provides algebraic tools for counting intersections, measuring multiplicities, and organizing geometric relations in a systematic way. The theory is essential in both enumerative geometry and the study of divisors.
9.1 Intersections of subvarieties
When subvarieties intersect, their meeting may be transverse, tangential, or embedded in a more complicated configuration. Intersection theory assigns algebraic meaning to these meetings, often turning geometric overlap into numerical data.
The resulting counts are stable under deformation in suitable settings. This makes intersection theory a powerful invariant of algebraic spaces.
9.2 Multiplicity
Multiplicity measures how strongly a subvariety or point appears in an intersection or solution set. A point where several branches meet may contribute with greater weight than a simple transverse intersection.
This concept refines naive counting and is crucial for obtaining correct algebraic formulas. It also reflects subtle local structure near singularities.
9.3 Chow rings
The Chow ring organizes algebraic cycles modulo rational equivalence, with multiplication given by intersection. It packages geometric classes into an algebraic structure that supports calculations and invariants.
Chow rings are fundamental in intersection theory because they encode how subvarieties combine. They also connect to cohomological and enumerative techniques.
9.4 Bézout's theorem
Bézout's theorem gives the expected number of intersections of projective plane curves under suitable conditions. More generally, it relates degrees of varieties to intersection numbers.
The theorem illustrates how algebraic degree controls geometric complexity. It is one of the most famous counting results in algebraic geometry.
10 Morphisms and mappings
Morphisms are the maps that preserve algebraic structure between varieties or schemes. They determine how spaces relate to one another and are central to concepts such as fibers, images, and equivalence. Different classes of morphisms capture different geometric behaviors.
10.1 Finite morphisms
A finite morphism is one that corresponds algebraically to a finite module extension. Such maps behave like finite coverings in many respects, though with algebraic rather than topological constraints.
Finite morphisms are important because they preserve strong finiteness properties and often arise in normalization and projection arguments. They also play a role in studying extension fields.
10.2 Proper morphisms
A proper morphism is an algebraic analogue of a compact map. It satisfies a valuative criterion that ensures good behavior under specialization and limits.
Properness is a key condition in global geometry because it supports many finiteness and existence theorems. Projective morphisms are standard examples of proper morphisms.
10.3 Flat morphisms
A flat morphism is one that preserves exactness after tensoring, reflecting uniform behavior of fibers. Flatness prevents sudden changes in dimension or algebraic structure in many situations.
This condition is central to deformation theory and family constructions. It expresses a strong form of compatibility between local and global data.
10.4 Étale morphisms
An étale morphism is a smooth-like map with no ramification in an algebraic sense. It often serves as a local isomorphism and is important in the study of covering spaces and fundamental groups.
Étale maps are a key part of modern algebraic geometry because they capture subtle local behavior while remaining algebraic. They also lead to étale cohomology, a major tool in arithmetic applications.
10.5 Branched coverings
A branched covering is a map that is locally a covering away from a branch locus, where the behavior becomes singular or ramified. In algebraic geometry, such maps are often studied through finite morphisms with special degeneration.
Branched coverings are useful for constructing new varieties from old ones and for understanding how maps fail to be locally trivial. They connect geometric branching with algebraic ramification.
11 Moduli and parameter spaces
Moduli theory studies families of geometric objects and the spaces that classify them. Rather than examining one variety at a time, it asks how entire collections of varieties, bundles, or curves vary in families. This leads to parameter spaces that encode classification in a geometric form.
11.1 Moduli problems
A moduli problem asks for a space or object that classifies geometric structures up to an equivalence relation. The aim is to encode all such objects in a single parameter space that reflects their deformations and symmetries.
Moduli problems can be subtle because objects may have automorphisms or may not fit neatly into a naïve space. Their study motivates many advanced ideas, including stacks and geometric invariant theory.
11.2 Hilbert schemes
Hilbert schemes parameterize closed subschemes with a fixed numerical profile, such as a given Hilbert polynomial. They provide a systematic way to study families of subvarieties inside a fixed ambient space.
These schemes are fundamental in modern moduli theory because they represent many geometric deformation problems. They also help organize the geometry of embedded varieties.
11.3 Moduli of curves
The moduli of curves concerns the classification of algebraic curves up to isomorphism. Such moduli spaces record how curves of a fixed genus vary and how degenerations occur.
This subject is important because curves are both manageable and richly structured. Their moduli spaces have deep connections to topology, arithmetic, and deformation theory.
11.4 Moduli spaces of vector bundles
Moduli spaces of vector bundles classify bundles over a fixed variety according to rank, degree, stability, and related conditions. These spaces capture geometric data that vary over a base variety and often have intricate structure.
Vector bundle moduli spaces are central in geometry because bundles appear in cohomology, representation theory, and gauge-theoretic analogues. Stability conditions are frequently used to obtain well-behaved parameter spaces.
12 Algebraic geometry over other fields
Algebraic geometry changes substantially when the base field or ring is altered. Working over the real numbers, finite fields, or non-algebraically closed fields introduces new phenomena and new methods. These settings are especially important in arithmetic applications.
12.1 Real algebraic geometry
Real algebraic geometry studies solution sets over the real numbers. Unlike the complex case, real varieties can have disconnected components, bounded regions, and order-theoretic features.
This area connects algebraic methods with semialgebraic geometry and real analysis. It often focuses on the existence and structure of real points rather than only complexified geometry.
12.2 Algebraic geometry over finite fields
Over finite fields, algebraic varieties have only finitely many rational points, but their geometry can still be highly nontrivial. Counting points, understanding reductions, and studying zeta functions are central themes.
This setting is especially significant in number theory and coding theory. It also motivates cohomological methods designed to handle arithmetic point counts.
12.3 Geometry over non-algebraically closed fields
When the base field is not algebraically closed, varieties may lack points or may split only after field extension. This introduces descent questions and subtle distinctions between geometric and arithmetic properties.
Such geometry requires careful attention to how objects behave after extending scalars. Rational points, forms, and Galois actions become important organizing ideas.
12.4 Arithmetic geometry
Arithmetic geometry combines algebraic geometry with number theory. It studies varieties defined over rings and fields that carry arithmetic information, such as the integers or number fields.
The field investigates rational points, Diophantine equations, reduction modulo primes, and cohomological tools adapted to arithmetic contexts. It has become one of the most influential areas in modern mathematics.