Intersection number
In mathematics, the intersection number is an invariant that quantifies how two or more geometric objects, such as submanifolds in a topological space or divisors on an algebraic variety, intersect, accounting for both the count of intersection points and their multiplicities or orientations.[1][2] It provides a way to measure intersections in a manner invariant under deformations or equivalences, generalizing the intuitive notion of crossing points for curves to higher-dimensional and more abstract settings.[1] In differential topology, the intersection number typically refers to the signed or mod-2 count of transverse intersection points between compact oriented submanifolds of complementary dimensions in a smooth manifold. For instance, if X and Z are closed submanifolds of an n-manifold Y with \dim X + \dim Z = n, and the inclusion map i: X \to Y is transverse to Z, the mod-2 intersection number I_2(X, Z) is the cardinality of X \cap Z modulo 2; this extends to signed integers when orientations are considered.[2] Key properties include homotopy invariance—meaning the number remains unchanged under homotopies of maps transverse to the submanifold—and the boundary theorem, which states that if X = \partial W for some manifold W and the map extends over W, then the intersection number vanishes.[2] These features make intersection numbers essential for computing topological invariants like the Euler characteristic via the Poincaré-Hopf theorem or for applications in cobordism theory.[2] In algebraic geometry, the intersection number for divisors D_1, \dots, D_n on a nonsingular n-dimensional variety V at a point P where they intersect properly is defined as the dimension of the local ring quotient \mathcal{O}_{V,P} / (f_1, \dots, f_n), where f_i are local equations for the D_i; the global intersection number is the sum over all such points.[1] For plane curves, this local multiplicity i(f, g; p) at a point p measures the "order of contact," as seen in Bézout's theorem, which equates the total intersection number of two projective curves of degrees d and e to de over an algebraically closed field, provided they share no common component.[3] Properties such as additivity in each divisor, invariance under linear equivalence, and compatibility with pullbacks under finite maps ensure that intersection numbers descend to the level of Chow groups or other intersection theory frameworks, playing a central role in enumerative geometry and the study of cycles.[1]Fundamentals
Basic Concept
The intersection number provides an algebraic measure of how two submanifolds or cycles intersect within a larger ambient space, generalizing the intuitive geometric count of intersection points by incorporating signs from orientations and adjustments for multiplicities in non-transverse cases. This count captures the topological linking or crossing between the objects, remaining well-defined even when direct geometric intersections are perturbed or absent, such as in the case of skew lines in three-dimensional space whose intersection number reflects their relative twisting.[4][5] A simple example occurs in the Euclidean plane, where two distinct non-parallel lines intersect transversely at a single point, yielding an intersection number of 1, while parallel lines do not intersect and thus have an intersection number of 0; this aligns with Bézout's theorem, which states that two curves of degrees d_1 and d_2 intersect in d_1 d_2 points counting multiplicity, here $1 \times 1 = 1. Such examples illustrate how the intersection number encodes essential geometric information algebraically, serving as a foundation for more abstract settings like homology pairings on manifolds.[4] This basic concept establishes the invariance of intersection numbers under continuous deformations of the submanifolds—intuitively, one can "move" the objects slightly without altering the count, as long as their topological types (homology classes) are preserved—making it a robust prerequisite for formal definitions in topology and algebraic geometry.[4] In fundamental cases, the intersection number exhibits bilinearity, being linear in each argument when fixing the other, and skew-symmetry, where swapping the two cycles negates the number, ensuring consistency with orientation conventions.[5]Historical Overview
The concept of the intersection number traces its origins to the late 19th century, where Henri Poincaré introduced intersection indices as part of his foundational work in topology. In his 1895 paper "Analysis Situs," Poincaré developed the idea of counting intersections between cycles to define topological invariants, laying the groundwork for what would become algebraic topology. This approach addressed the need for invariant quantities in the study of manifolds, motivated by problems in differential equations and global geometry during the 1880s.[6] The early 20th century saw significant formalization in algebraic topology, particularly through Solomon Lefschetz's contributions in the 1920s. Lefschetz extended Poincaré's ideas by developing intersection theory for manifolds and simplicial complexes, introducing the intersection pairing in homology groups and applying it to fixed-point theorems. His work, detailed in publications like "L'analysis situs et la géométrie algébrique" (1924), bridged topology and algebraic geometry, enabling the study of cycles on projective varieties. By the 1940s, André Weil advanced the theory in algebraic geometry with his axiomatic treatment of intersection multiplicities for cycles on varieties over algebraically closed fields, as presented in "Foundations of Algebraic Geometry" (1946). Weil's framework emphasized rigorous definitions independent of coordinates, resolving ambiguities in classical enumerative problems. Key milestones in the mid-20th century included W.V.D. Hodge's integration of intersection theory with differential geometry and characteristic classes during the 1950s. Hodge's index theorem, linking intersection numbers on algebraic varieties to topological invariants via harmonic forms, connected the theory to broader analytic tools and foreshadowed index theorems in operator theory. This period also featured the Riemann-Roch theorem's evolution from its 19th-century origins—proved by Bernhard Riemann in 1857 and Gustav Roch in 1865 for curves, motivating intersection counts as dimensions of linear systems—to 20th-century generalizations like Hirzebruch's 1956 Riemann-Roch theorem, which incorporated Todd classes for higher-dimensional varieties. The culmination of these developments appeared in William Fulton's comprehensive "Intersection Theory" (1984), which unified topological and algebraic approaches through refined notions of refined intersections and excess bundles, establishing a modern foundation applicable across geometry. In recent decades, intersection theory has extended into derived geometry, addressing non-transverse intersections via derived categories and stacks. Andrei Căldăraru's post-2010 work, such as his 2013 paper on derived intersections and the algebraic Hodge theorem, demonstrates formality results for derived intersections of smooth subvarieties, enabling generalizations of classical theorems to singular or non-reduced settings and linking to mirror symmetry applications. These advancements, building on Fulton's framework, continue to influence enumerative geometry and physics-inspired problems as of 2025.[7]Topological Definitions
Manifolds and Cycles
In the topological framework for intersection numbers, the setting begins with smooth manifolds and their submanifolds, where homology is computed using singular chains. A smooth manifold M of dimension n is equipped with an atlas of charts to local Euclidean spaces, allowing the definition of smooth maps and tangent spaces. Submanifolds are smoothly embedded copies of lower-dimensional manifolds within M, and singular homology H_*(M; \mathbb{Z}) is generated by singular chains, which are formal integer linear combinations of continuous maps from standard simplices into M.[4] Cycles in this context are closed singular chains, meaning elements of the kernel of the boundary operator in the chain complex, representing homology classes in H_p(M; \mathbb{Z}). Boundaries, which are images of the boundary operator from higher-dimensional chains, intersect trivially with all cycles due to the properties of the chain complex, ensuring that intersection numbers depend only on homology classes rather than specific chain representatives.[4] Transversality provides the condition for well-defined intersections between submanifolds or maps representing cycles. Two smooth submanifolds X \subset M of dimension p and Y \subset M of dimension q with p + q = n intersect transversely if at every intersection point, their tangent spaces span the tangent space of M. For generic choices of representatives, transversality can be achieved via smooth perturbations, as any two cycles can be approximated by transverse ones without altering their homology classes, leveraging the density of transverse maps in the space of smooth embeddings.[8] The basic intersection number for transverse cycles \alpha \in H_p(M) and \beta \in H_q(M) with p + q = n is defined as the signed algebraic count of their intersection points in M. Each transverse intersection point contributes +1 or -1 depending on whether the orientations induced by \alpha and \beta agree with the orientation of M, yielding an integer invariant of the homology classes. This count is independent of the choice of transverse representatives, as homotopies preserve the total signed number through boundary cancellations.[8] Poincaré-Lefschetz duality elevates this geometric intersection to an algebraic pairing on homology groups. For a compact oriented n-manifold M (possibly with boundary), the duality isomorphism H_p(M; \mathbb{Z}) \cong H^{n-p}(M; \mathbb{Z}) identifies the intersection number as a nondegenerate bilinear pairing H_p(M; \mathbb{Z}) \times H_{n-p}(M; \mathbb{Z}) \to \mathbb{Z}, where the pairing is given by the cap product with the fundamental class [M] composed with evaluation on the fundamental class. This framework ensures the intersection number captures essential topological invariants, such as in the study of manifold embeddings and duality theorems.[4]Homology Intersection Numbers
In singular homology with integer coefficients, the homology groups H_*(M; \mathbb{Z}) of a compact, connected, oriented n-manifold M are finitely generated abelian groups, with Poincaré duality providing an isomorphism H_k(M; \mathbb{Z}) \cong H^{n-k}(M; \mathbb{Z}) for each k, where the cohomology groups are computed using cochains dual to chains via the manifold's orientation.[4] This duality arises from the cap product with the fundamental class [M] \in H_n(M; \mathbb{Z}), which is the generator corresponding to the orientation, ensuring that homology classes can be paired algebraically while respecting the manifold's topological structure.[4] The intersection product in homology is defined algebraically as a nondegenerate bilinear form H_k(M; \mathbb{Z}) \times H_{n-k}(M; \mathbb{Z}) \to \mathbb{Z}, often denoted \langle \alpha \cdot \beta \rangle for classes \alpha \in H_k(M; \mathbb{Z}) and \beta \in H_{n-k}(M; \mathbb{Z}). Geometrically, this number is computed by representing \alpha and \beta with cycles (singular chains with zero boundary) that are made transverse via small perturbations, yielding a signed count \sum \varepsilon_i over the finite intersection points, where each \varepsilon_i = \pm 1 is determined by the local orientations at the point: positive if the orientations of the cycles and ambient manifold align consistently, and negative otherwise.[4] Algebraically, this pairing can be expressed via Poincaré duality as \alpha \cdot \beta = \langle \mathrm{PD}(\alpha), \beta \rangle, with \mathrm{PD} the duality isomorphism. The form satisfies graded commutativity \alpha \cdot \beta = (-1)^{k(n-k)} \beta \cdot \alpha.[4] This intersection number is invariant under homotopy of the representing cycles, as the general position theorem (or moving lemma) guarantees that any two cycles can be homotoped to transverse position without altering their homology classes, preserving the algebraic count of intersections; a proof involves excising small neighborhoods around intersection points and adjusting via boundaries, but the details rely on the triangulable nature of manifolds.[4] For example, on the 2-torus T^2 = S^1 \times S^1, the standard generators of H_1(T^2; \mathbb{Z}) \cong \mathbb{Z} \oplus \mathbb{Z} are the meridian \mu (a loop around one factor) and longitude \lambda (around the other); these can be realized as transverse curves intersecting at a single point with positive sign, yielding intersection number +1, while \mu \cdot \mu = 0 and \lambda \cdot \lambda = 0 since self-intersections can be resolved to zero algebraic count.[4] More generally, for classes (p,q) and (r,s) in this basis, the intersection is the determinant ps - qr, reflecting the symplectic structure induced by the pairing.[4] For non-compact oriented manifolds, the intersection number extends to classes represented by proper cycles (maps from compact domains that are proper, ensuring preimages of compact sets are compact), where transversality yields compact intersection sets, and the signed count remains well-defined as a topological invariant, often computed using compactly supported cohomology to handle infinite extent.[4] This requires the cycles to intersect properly, meaning their supports do not escape to infinity without bound, preserving the finiteness of the [sum \sum](/page/Sum_Sum) \varepsilon_i.[4]Algebraic Definitions
Riemann Surfaces
On a compact Riemann surface X, a divisor is a formal finite sum D = \sum_{p \in X} n_p p, where n_p \in \mathbb{Z} are the multiplicities (or orders) at points p, with all but finitely many n_p = 0. The group of all divisors forms \operatorname{Div}(X), and the degree of D is \deg D = \sum_p n_p.[9] The intersection pairing between two divisors D, E \in \operatorname{Div}(X) is defined algebraically by (D \cdot E) = \sum_{p \in X} \operatorname{mult}_p(D) \operatorname{mult}_p(E), where the sum counts the product of multiplicities at each point p. This bilinear form measures the "overlap" of supports with multiplicities. In the context of the Riemann-Roch theorem, which states that for a divisor D, \dim L(D) - \dim L(K - D) = \deg D - g + 1 (where L(D) is the space of meromorphic functions with poles bounded by D, K is a canonical divisor with \deg K = 2g - 2, and g is the genus), the pairing requires adjustment for base points of the associated linear systems |D| and |E|. Base points, common to all effective divisors in a linear system, contribute fixed multiplicities to the pairing; the adjustment subtracts or accounts for these fixed components to compute the moving part's intersections, ensuring compatibility with dimension formulas for L(D + E).[10][9] Via Serre duality on X, which pairs H^0(X, \mathcal{O}_X(D))^\vee \cong H^1(X, \omega_X \otimes \mathcal{O}_X(-D)) (with \omega_X = \mathcal{O}_X(K) the canonical sheaf), the trace map induces a nondegenerate pairing on cohomology that aligns the algebraic intersection with analytic residues.[1] For example, on an elliptic curve (genus g=1), consider two effective divisors of degree 1 given by distinct points P and Q. Their intersection number is 0. The divisor P + Q has degree 2, and by the Riemann-Roch theorem, \dim L(P + Q) = 2, which embeds the curve via the complete linear system |P + Q|.[10] This algebraic definition bridges to topology: the uniformization theorem identifies X with a quotient of the hyperbolic plane (for g \geq 2) or other models, preserving the first homology group H_1(X, \mathbb{Z}); the algebraic pairing on divisors corresponds to the topological intersection pairing on 1-cycles via the Abel-Jacobi map sending points to homology classes.[11]Algebraic Varieties
In algebraic geometry, the intersection number for algebraic varieties extends the concept to higher-dimensional settings over algebraically closed fields, utilizing cycles and their equivalence classes in Chow groups. A cycle on a variety X is a formal \mathbb{Z}-linear combination of irreducible subvarieties of X, and the Chow group A_k(X) consists of k-dimensional cycles modulo rational equivalence, where two cycles are rationally equivalent if their difference is a \mathbb{Z}-linear combination of principal divisors on subvarieties of dimension k+1. Rational equivalence is generated by cycles of the form \operatorname{div}(f), the divisor of a rational function f on an integral subvariety, ensuring that the Chow groups capture algebro-geometric analogs of homology. The intersection product in the Chow groups A_*(X) is defined for cycles on a projective variety X by applying the moving lemma to displace one cycle to a rationally equivalent position where it intersects the other properly, followed by taking the cycle class of the scheme-theoretic intersection components.[12] This product equips the direct sum \bigoplus_k A_k(X) with a ring structure, known as the Chow ring, when X is smooth. A proper intersection occurs when two subvarieties D and E in X satisfy \operatorname{codim}(D) + \operatorname{codim}(E) = \operatorname{codim}(D \cap E), which dimensionally translates to \dim(D \cap E) = \dim D + \dim E - \dim X.[13] For transverse intersections, where the intersection is a union of components of the expected dimension without excess structure, the intersection class is the sum of the classes of these components, each weighted by its multiplicity as a cycle. In the specific case of two curves of degrees d_1 and d_2 in the projective plane \mathbb{P}^2, Bézout's theorem asserts that their intersection number is d_1 d_2, counting points with appropriate multiplicities under proper intersection conditions.[14] To handle cases where intersections do not yield integer classes, the Chow groups are often tensored with \mathbb{Q} to form A_*(X) \otimes \mathbb{Q}, enabling the intersection product to produce fractional coefficients and facilitating computations in the rational Chow ring.[12] This extension preserves the ring structure and is essential for enumerative problems where rational multiples arise naturally.Multiplicities and Formulas
Plane Curve Multiplicities
In the local setting, the intersection multiplicity of two plane algebraic curves defined by polynomials f(x,y) = 0 and g(x,y) = 0 in \mathbb{C}^2 is considered at a point p = (0,0) where both curves pass through the origin.[15] This setup captures the local behavior near the intersection point, assuming the curves are defined over the complex numbers for analytic properties.[16] The intersection multiplicity i_p(f,g) at p is defined as the dimension over \mathbb{C} of the quotient ring \mathbb{C}[[x,y]] / (f,g), where \mathbb{C}[[x,y]] is the ring of formal power series in two variables.[15] This length measures the "order of contact" between the curves at p, yielding 1 for transverse intersections and higher values when the curves are tangent or more singularly coincident, assuming no common component through p.[16] To compute i_p(f,g), one method uses the resultant \operatorname{Res}(f,g,t) treated as a polynomial in one variable after homogenizing or substituting, where the multiplicity equals the order of vanishing at the origin.[17] Alternatively, the resultant can be obtained via the determinant of the Sylvester matrix associated to f and g, providing an explicit algebraic tool for calculation.[17] For example, consider the line g(x,y) = y = 0 (the x-axis) and the parabola f(x,y) = y - x^2 = 0 at the origin. Here, i_{(0,0)}(f,g) = 2, reflecting the tangency as the parabola touches the line.[18] Key properties include additivity: if h is another polynomial with no common component with f at p, then i_p(f,gh) = i_p(f,g) + i_p(f,h), allowing decomposition of intersections.[16] Additionally, the multiplicity is continuous under small deformations of the curves, remaining constant as long as the intersection point persists without common components.[15] The sum of these local multiplicities over all points yields the global intersection number, as in Bézout's theorem.[16]Serre's Tor Formula
In the sheaf-theoretic framework, Serre defined the local intersection multiplicity at a point p for two subschemes X and Y of a scheme Z, with supports contained in p, asi_p(X, Y) = \sum_{i \geq 0} (-1)^i \length_{\mathcal{O}_{Z,p}} \left( \Tor_i^{\mathcal{O}_{Z,p}} (\mathcal{O}_X, \mathcal{O}_Y) \right).
This formulation arises in the context of coherent sheaves on Noetherian schemes, where \mathcal{O}_X and \mathcal{O}_Y are the structure sheaves of the subschemes, and the Tor functor measures the failure of the tensor product to be exact. For effective cycles on a variety, the global intersection number extends this local definition by summing over the components of the intersection: if \alpha = \sum n_i [V_i] and \beta = \sum m_j [W_j] are cycles with proper intersection supported on components Z_k, then the intersection cycle is \alpha \cdot \beta = \sum_k e_k [Z_k], where the coefficient e_k = \sum_{i,j : V_i \cap W_j = Z_k} n_i m_j i_{Z_k}(V_i, W_j), aggregating the local lengths weighted by cycle multiplicities.[19] A proof of this formula relies on resolving one of the structure sheaves via the Koszul complex associated to a regular sequence generating the ideal sheaf; the homology of the tensor product with the other structure sheaf then computes the Tor groups, with the alternating sum of lengths yielding the multiplicity, and in cases where higher Tor terms vanish for i > 1, reducing to \length(\Tor_0) - \length(\Tor_1). This approach offers key advantages over classical definitions, as it naturally accommodates non-reduced subschemes—where ideal sheaves may not be generated by regular sequences—and extends seamlessly to intersections in higher codimensions without requiring transversality assumptions.[20] A representative example is the intersection of two planes in \mathbb{A}^3 that meet along a line, then intersected with a third plane transverse to the line at a point p; the non-transverse configuration yields \Tor_1^{\mathcal{O}_{\mathbb{A}^3,p}}(\mathcal{O}_X, \mathcal{O}_Y) of length 2, capturing the double point multiplicity.[21]