Intersection theory is a powerful tool in algebraic geometry, studying how subspaces intersect and computing associated invariants. It connects geometry, topology, and algebra, generalizing classical results like Bézout's theorem to higher dimensions and abstract settings.
This theory quantifies intersections using intersection numbers, which are invariant under continuous deformations. The , a bilinear operation on Chow groups or cohomology, encodes these intersections and is compatible with pullbacks and pushforwards, making it invaluable for computations.
Intersection theory overview
Intersection theory studies how subspaces of a geometric object intersect and provides tools to compute invariants associated to these intersections
Plays a central role in modern algebraic geometry, connecting geometry, topology, and algebra
Generalizes classical results like Bézout's theorem and the Riemann-Roch theorem to higher dimensions and more abstract settings
Motivation for intersection theory
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Need to understand how subvarieties of algebraic varieties intersect and behave under operations like intersections and unions
Intersections carry important geometric and topological information (number of intersection points, multiplicity, etc.)
Intersection theory provides a rigorous framework to study and compute these invariants
Topological vs algebraic intersections
Topological intersections consider the underlying topological spaces and how subspaces intersect as point sets
Algebraic intersections take into account the scheme structure and multiplicities of intersection points
Algebraic intersections are more refined and carry additional information compared to topological intersections
Transversality of intersections
Transversal intersections occur when subvarieties meet "nicely" without tangencies or higher-order contact
Transversality ensures well-defined intersection multiplicities and simplifies computations
Generic intersections are often transversal, but special techniques are needed for non-transversal cases (excess intersections, blow-ups, etc.)
Intersection numbers
Intersection numbers quantify the intersection of subvarieties in a compact oriented manifold or a projective variety
Assign a numerical invariant to the intersection, taking into account multiplicities and orientations
Intersection numbers are invariant under continuous deformations and algebraic equivalence
Definition of intersection numbers
For subvarieties X and Y of complementary in a compact oriented manifold M, the intersection number X⋅Y is defined as the signed count of intersection points, with signs determined by orientations
In the algebraic setting, intersection numbers are defined using the cup product in cohomology or the Chow ring of the variety
Properties of intersection numbers
Intersection numbers are bilinear and symmetric
Satisfy a projection formula relating intersections with pullbacks and pushforwards
Intersection numbers are invariant under rational equivalence of cycles
Intersection numbers of subvarieties
For subvarieties X and Y of a projective variety V, the intersection number X⋅Y is the degree of the cycle class [X]∩[Y] in the Chow ring of V
Intersection numbers can be computed using the cup product in cohomology via
Examples: intersection number of a curve and a hypersurface in Pn, of a divisor on a surface
Intersection product
The intersection product is a bilinear operation on the Chow groups or cohomology of a variety that encodes the intersection of cycles or cohomology classes
Generalizes the cup product in cohomology and the intersection of subvarieties
Intersection product is compatible with pullbacks and pushforwards, making it a powerful tool for computations
Cup product in cohomology
The cup product is a bilinear operation on the cohomology of a topological space or a variety
For cohomology classes α∈Hp(X) and β∈Hq(X), their cup product α∪β is a class in Hp+q(X)
Cup product is associative, graded-commutative, and compatible with pullbacks
Cap product with homology
The cap product is a bilinear pairing between cohomology and homology, producing a homology class
For a α∈Hp(X) and a homology class σ∈Hq(X), their cap product α∩σ is a class in Hq−p(X)
Cap product is related to Poincaré duality and allows for computations involving both cohomology and homology
Poincaré duality and intersections
Poincaré duality establishes an isomorphism between the cohomology and homology of a compact oriented manifold, with degree shifted by the
Under Poincaré duality, the intersection product of submanifolds corresponds to the cup product of their dual cohomology classes
Poincaré duality allows for the computation of intersection numbers using cohomological methods
Chow groups
Chow groups are algebraic analogues of homology groups for algebraic varieties
Elements of the Chow group Ak(X) are formal linear combinations of k-dimensional subvarieties of X, modulo rational equivalence
Chow groups carry a natural intersection product, making them a powerful tool in intersection theory
Construction of Chow groups
Start with the free abelian group generated by k-dimensional subvarieties of a variety X
Quotient by the subgroup generated by divisors of rational functions on subvarieties of dimension k+1
The resulting quotient group is the Chow group Ak(X), with elements called cycle classes
Intersection product on Chow groups
The intersection product on Chow groups is induced by the geometric intersection of subvarieties
For cycle classes α∈Ak(X) and β∈Al(X), their intersection product α⋅β is a class in Ak+l−n(X), where n is the dimension of X
Intersection product on Chow groups satisfies properties like bilinearity, commutativity, and associativity
Examples of Chow groups
For a smooth projective curve C, the Chow group A0(C) is isomorphic to Z, generated by the class of a point
For a smooth projective surface S, the Chow group A1(S) is the Picard group of S, classifying divisors modulo linear equivalence
Chow groups of projective spaces Pn are isomorphic to Z[H]/(Hn+1), where H is the class of a hyperplane
Riemann-Roch theorem
The Riemann-Roch theorem is a fundamental result in algebraic geometry relating the geometry of a curve or variety to its algebraic properties
Provides a formula for the dimension of the space of global sections of a line bundle or a divisor in terms of intersection numbers and characteristic classes
Generalizations of the Riemann-Roch theorem, such as the Hirzebruch-Riemann-Roch theorem and the Grothendieck-Riemann-Roch theorem, play a central role in intersection theory
Statement of Riemann-Roch theorem
For a smooth projective curve C and a divisor D on C, the Riemann-Roch theorem states:
dimH0(C,O(D))−dimH1(C,O(D))=deg(D)−g+1
where g is the genus of C
For a smooth projective variety X and a line bundle L on X, the Hirzebruch-Riemann-Roch theorem expresses the Euler characteristic of L in terms of Chern classes and the Todd class of X
Proof sketch of Riemann-Roch
The proof of the Riemann-Roch theorem for curves relies on the study of the canonical divisor and the Serre duality theorem
Key steps include analyzing the degree of the canonical divisor, applying Serre duality to relate cohomology groups, and using the properties of the Euler characteristic
Proofs of generalizations like the Hirzebruch-Riemann-Roch theorem involve characteristic classes and the Grothendieck group of coherent sheaves
Applications of Riemann-Roch
Computing dimensions of linear systems and spaces of global sections of line bundles
Studying the geometry of curves and surfaces, such as the existence of special divisors or embeddings into projective spaces
Proving the Kodaira vanishing theorem and its generalizations, which relate positivity of line bundles to the vanishing of cohomology groups
Excess intersections
Excess intersection theory deals with the situation when the intersection of subvarieties has a higher dimension than expected
Occurs when the subvarieties intersect non-transversally or have a common component
Excess intersection formulas express the intersection product in terms of Chern classes and Segre classes of the normal bundles to the intersection
Excess intersection formula
For subvarieties X and Y of a variety V, with intersection Z=X∩Y of excess dimension e, the excess intersection formula states:
[X]⋅[Y]=j∗(α)
where j:Z→V is the inclusion, and α is a cycle class on Z involving the Chern classes of the normal bundles NX/V and NY/V
The precise formula for α depends on the context and the type of excess intersection (Fulton-MacPherson, Vogel, etc.)
Computations with excess intersections
Excess intersection formulas allow for the computation of intersection products in non-transversal situations
Involve calculating Chern classes and Segre classes of normal bundles, often using the splitting principle or the Grothendieck-Riemann-Roch theorem
Examples: intersections of divisors on a surface, intersections of subvarieties in a projective bundle
Generalizations of excess intersections
Refined intersection products, such as the Fulton-MacPherson intersection product, take into account higher-order tangencies and provide more detailed intersection information
Excess intersection theory can be formulated in the context of derived algebraic geometry, using derived schemes and
Excess intersections play a role in enumerative geometry, quantum cohomology, and the study of moduli spaces
Intersection theory on singular varieties
Intersection theory can be extended to singular varieties, where the usual definitions of intersections and multiplicities may not apply directly
Requires working with more general objects like schemes, coherent sheaves, and K-theory
Different approaches to intersection theory on singular varieties have been developed, each with its own advantages and challenges
Intersection theory on schemes
Schemes provide a general framework for studying singular varieties and their intersections
Intersection theory on schemes involves the study of Chow groups, Chern classes, and Segre classes of coherent sheaves
Key tools include the Grothendieck group of coherent sheaves, the Grothendieck-Riemann-Roch theorem, and the Fulton-MacPherson intersection product
Fulton's intersection theory
William Fulton developed a comprehensive approach to intersection theory on singular varieties using the language of schemes and coherent sheaves
Fulton's intersection theory is based on the construction of the Chow group of a scheme and the definition of the intersection product using the Segre class of the normal cone
Provides a unified framework for studying intersections, excess intersections, and characteristic classes on singular varieties
Intersection theory vs K-theory
K-theory is another approach to studying algebraic cycles and vector bundles on varieties, based on the Grothendieck group of vector bundles
Intersection theory and K-theory are closely related, with the Chern character providing a connection between the two theories
Some results in intersection theory, such as the Hirzebruch-Riemann-Roch theorem, have natural analogues in K-theory, like the Grothendieck-Riemann-Roch theorem
Key Terms to Review (18)
Alexander Grothendieck: Alexander Grothendieck was a renowned French mathematician who made groundbreaking contributions to algebraic geometry, homological algebra, and category theory. His work revolutionized the way these fields were understood, particularly through his development of concepts such as sheaves, schemes, and cohomology theories, connecting various mathematical areas and providing deep insights into their structure.
Čech Cohomology: Čech cohomology is a type of cohomology theory that is used in algebraic topology to study the properties of topological spaces through the use of open covers. It focuses on the relationships between local and global properties of these spaces, providing a powerful tool to analyze them using the language of sheaves and derived functors.
Classical intersection theory: Classical intersection theory is a mathematical framework used to study how subvarieties intersect within a given geometric space, often employing tools from algebraic geometry. It provides techniques to calculate intersection numbers, which help in understanding the properties and dimensions of these intersections. This theory is foundational for many results in algebraic geometry and relates closely to concepts such as cohomology and algebraic cycles.
Codimension: Codimension is a mathematical concept that measures the difference between the dimension of a space and the dimension of a subspace within it. In simple terms, if you have a subspace embedded in a larger space, the codimension tells you how many dimensions are 'missing' from that larger space to fully accommodate the subspace. This idea is particularly important in intersection theory, where understanding the dimensions of intersecting spaces can lead to deeper insights about their properties and relationships.
Cohomology Class: A cohomology class is an equivalence class of cochains that represent a particular cohomology group, providing a way to capture topological information about a space. Each class contains cochains that yield the same cohomology group when applied to certain chains, highlighting essential features such as continuity and boundaries in geometric structures. Cohomology classes are integral to understanding intersection theory, as they help categorize how different subspaces intersect and relate to the overall topology of the space.
Degree of intersection: The degree of intersection refers to the number of points at which two geometric objects, such as curves or surfaces, intersect. It plays a crucial role in intersection theory, as it quantifies how these objects relate to one another in terms of their positions and intersections, which can impact various mathematical and geometrical concepts, including cohomology and algebraic geometry.
Derived Categories: Derived categories are a mathematical framework used in homological algebra to systematically study complexes of objects and their morphisms. They help in understanding the relationships between different mathematical structures by focusing on the derived functors, which allow one to capture more information than is possible with just the original objects. This concept is particularly useful in fields like algebraic geometry, where it facilitates the exploration of intersection theory and connecting homomorphisms.
Dimension: In mathematics, dimension refers to the minimum number of coordinates needed to specify a point within a given space. It connects to various concepts, such as the geometric structure of spaces and the algebraic properties of objects, which are essential for understanding relationships in areas like topology and algebraic geometry.
Fundamental group: The fundamental group is a mathematical concept in algebraic topology that captures the idea of 'loops' in a space and their equivalence classes under continuous deformation. It is defined for a given topological space based on a specific base point and provides information about the space's shape by identifying how many distinct loops can be formed based on the paths that can be continuously transformed into one another. This group is vital in understanding the properties of spaces, especially in intersection theory where it helps in analyzing how different subspaces intersect.
Henri Poincaré: Henri Poincaré was a French mathematician, theoretical physicist, and philosopher of science, known for his foundational contributions to topology, dynamical systems, and the philosophy of mathematics. His work laid important groundwork for the development of modern topology and homology theory, influencing how mathematicians understand spaces and their properties.
Homotopy: Homotopy is a concept in topology that describes a continuous deformation between two continuous functions. Essentially, if one function can be transformed into another through a series of continuous changes without breaking, they are considered homotopic. This notion is crucial in understanding how spaces can be manipulated and related, especially when considering the properties of spaces and how they connect to algebraic structures.
Intersection product: The intersection product is a fundamental operation in algebraic topology that combines cohomology classes to produce new cohomology classes, representing the geometric intersection of submanifolds in a manifold. This operation provides a way to study how different cycles intersect and interact, linking the topology of the space with algebraic structures. It plays a crucial role in understanding various topological properties and invariants of manifolds.
Künneth Formula: The Künneth Formula is a powerful result in algebraic topology that describes how the homology or cohomology groups of the product of two topological spaces relate to the homology or cohomology groups of the individual spaces. It provides a way to compute the homology or cohomology of a product space based on the known properties of its components, connecting directly to various aspects of algebraic topology, including operations and duality.
Poincaré Duality: Poincaré Duality is a fundamental theorem in algebraic topology that establishes a relationship between the cohomology groups of a manifold and its homology groups, particularly in the context of closed oriented manifolds. This duality implies that the k-th cohomology group of a manifold is isomorphic to the (n-k)-th homology group, where n is the dimension of the manifold, revealing deep connections between these two areas of topology.
Proper intersection: Proper intersection refers to the scenario where two geometric objects intersect transversely, meaning their intersection has the expected dimension based on their dimensions and position in space. This concept is fundamental in understanding how shapes interact in intersection theory, revealing important relationships between their topological properties and dimensions.
Self-intersection: Self-intersection refers to the phenomenon where a geometric object intersects itself, creating overlapping points within the object itself. This concept is crucial in intersection theory as it allows for the study of how shapes interact with themselves and provides insights into their topological properties and algebraic characteristics.
Sheaf Cohomology: Sheaf cohomology is a mathematical tool used to study the global properties of sheaves on topological spaces through the use of cohomological techniques. It allows for the calculation of the cohomology groups of a sheaf, providing insights into how local data can give rise to global information, which connects with several important concepts in algebraic topology and algebraic geometry.
Transversal Intersection: Transversal intersection refers to the scenario where two or more geometric objects intersect in a way that is transverse, meaning their intersection is non-degenerate. This concept is crucial in intersection theory, as it allows for a clear understanding of how objects relate to one another through their points of intersection, leading to significant implications for topology and algebraic geometry.