11.1 Intersection theory

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 intersection product, 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 codimension in a compact oriented manifold $M$, the intersection number $X \cdot Y$ 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 $X \cdot Y$ is the degree of the cycle class $[X] \cap [Y]$ in the Chow ring of $V$
• Intersection numbers can be computed using the cup product in cohomology via Poincaré duality
• Examples: intersection number of a curve and a hypersurface in $\mathbb{P}^n$, self-intersection 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 $\alpha \in H^p(X)$ and $\beta \in H^q(X)$, their cup product $\alpha \cup \beta$ is a class in $H^{p+q}(X)$
• 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 cohomology class $\alpha \in H^p(X)$ and a homology class $\sigma \in H_q(X)$, their cap product $\alpha \cap \sigma$ is a class in $H_{q-p}(X)$
• 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 dimension
• 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 $A_k(X)$ 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 $A_k(X)$, 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 $\alpha \in A_k(X)$ and $\beta \in A_l(X)$, their intersection product $\alpha \cdot \beta$ is a class in $A_{k+l-n}(X)$, 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 $A_0(C)$ is isomorphic to $\mathbb{Z}$, generated by the class of a point
• For a smooth projective surface $S$, the Chow group $A_1(S)$ is the Picard group of $S$, classifying divisors modulo linear equivalence
• Chow groups of projective spaces $\mathbb{P}^n$ are isomorphic to $\mathbb{Z}[H]/(H^{n+1})$, 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: $\dim H^0(C, \mathcal{O}(D)) - \dim H^1(C, \mathcal{O}(D)) = \deg(D) - g + 1$ where $g$ is the genus of $C$
• For a smooth projective variety $X$ and a line bundle $\mathcal{L}$ on $X$, the Hirzebruch-Riemann-Roch theorem expresses the Euler characteristic of $\mathcal{L}$ 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 $Z = X \cap Y$ of excess dimension $e$, the excess intersection formula states: $[X] \cdot [Y] = j_*(\alpha)$ where $j: Z \to V$ is the inclusion, and $\alpha$ is a cycle class on $Z$ involving the Chern classes of the normal bundles $N_{X/V}$ and $N_{Y/V}$
• The precise formula for $\alpha$ 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 derived categories
• 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