04.18.01 · algebraic-geometry / intersection-theory

Intersection theory — Chow rings and enumerative geometry

shipped3 tiersLean: none

Anchor (Master): Fulton 1984/1998 Intersection Theory (Springer); Eisenbud-Harris 2016 — full proofs of the intersection product via reduction to the diagonal, deformation to the normal cone, and the excess intersection formula.

Intuition Beginner

When two curves in the plane cross, they meet in points. Intersection theory answers two questions about such a meeting: how many points are there, and how should each one be counted when the curves touch tangentially or overlap along a piece? The cleanest answer in mathematics — Bézout's theorem, that two plane curves of degrees $m$ and $n$ meet in $m \cdot n$ points — lives in a piece of algebra called the Chow ring.

The big idea is to replace each subvariety by an algebraic object recording its "class," and to define the meeting of two subvarieties as the product of their classes. The class of a curve in the plane is its degree; the class of a surface in space is its degree; and the intersection of two classes is a number you compute by multiplication. The Chow ring is the home of these classes and the place where this multiplication is honest.

Why does the counting need care? Two subvarieties may overlap along a whole piece rather than meet in isolated points, and tangencies must be weighted correctly. The modern theory — built by Chow, Grothendieck, Fulton, and MacPherson — makes the counting rigorous on any smooth variety. The reward is enumerative geometry: classical puzzles such as the 27 lines on a smooth cubic surface, or the 3264 conics tangent to five given conics, become clean computations in a ring.

Visual Beginner

Two plane curves in the projective plane, crossing at several marked points. Each crossing carries a small positive integer — the local intersection multiplicity — and the weighted sum of the multiplicities is the total intersection number.

Worked example Beginner

Bézout's theorem is the prototype of intersection counting. Two plane curves in the projective plane, of degrees $m$ and $n$ , with no shared piece, meet in exactly $m \cdot n$ points when each is counted with its local intersection multiplicity.

Concretely: a line (degree 1) and a conic (degree 2) meet in $1 \cdot 2 = 2$ points. Two conics (both degree 2) meet in $2 \cdot 2 = 4$ points. A line (degree 1) and a cubic (degree 3) meet in $1 \cdot 3 = 3$ points. Two cubics meet in $3 \cdot 3 = 9$ points. A cubic and a quartic meet in $3 \cdot 4 = 12$ points.

The multiplicities are what make the count rigid. A line tangent to a conic touches it at a single point but with multiplicity 2, so the count is still 2. If a line is actually one of the two pieces of a reducible conic, the "no shared piece" hypothesis fails, and the theorem does not apply directly — which is exactly why that hypothesis is required.

Check your understanding Beginner

Exercise (medium, multiple choice).

Two plane curves share a common line as a component. What does Bézout's theorem say about their intersection?

A. They meet in finitely many points, the count being the product of degrees B. The "no common component" hypothesis fails, so Bézout's theorem does not apply directly; the shared component must be removed first C. They meet in exactly one point D. They cannot meet at all

Hint

Bézout's count of $mn$ points requires the intersection to be proper — the curves must not share a curve.

Answer

B. When two curves share a component, they intersect along an entire curve, not in a finite set of points, so the intersection is not proper. The statement of Bézout requires no common component precisely to rule out this situation. Once the shared component is factored out, a refined count applies to the remaining pieces.

Formal definition Intermediate+

Let $X$ be an algebraic scheme over a field $k$ . An algebraic $k$ -cycle on $X$ is a finite formal $Z$ -linear combination of integral closed subschemes of dimension $k$ :

Z = V \sum n_{V} [V], n_{V} \in Z .

The free abelian group of $k$ -cycles is denoted $Z_{k} (X)$ . A cycle is effective if every $n_{V} \geq 0$ .

Rational equivalence. Two $k$ -cycles $Z_{0}, Z_{1}$ are rationally equivalent (written $Z_{0} \sim_{rat} Z_{1}$ ) if there is a $(k + 1)$ -cycle $W$ on $X \times P^{1}$ , flat over $P^{1}$ , whose fibres over $0, \infty \in P^{1}$ are $Z_{0}$ and $Z_{1}$ . Equivalently, $Z_{0}$ can be deformed into $Z_{1}$ through an algebraic family parametrised by $P^{1}$ . Rational equivalence is generated by the relation $Z_{0} - Z_{1} = 0$ for $[k (P^{1})]$ -families of this form. This is the algebraic analogue of homotopy: cycles in the same rational-equivalence class play the same enumerative role.

Chow groups. The $k$ -th Chow group of $X$ is the group of $k$ -cycles modulo rational equivalence:

A_{k} (X) := Z_{k} (X) / \sim_{rat} .

When $X$ is equidimensional of dimension $n$ , it is often cleaner to grade by codimension: $A^{i} (X) := A_{n - i} (X)$ . So $A^{0} (X) = Z \cdot [X]$ , $A^{1} (X)$ is the group of Weil divisors modulo rational equivalence — on a smooth variety, the divisor class group $Cl (X)$ 04.05.01 — and $A^{n} (X)$ is the group of 0-cycles modulo rational equivalence.

Proper pushforward. If $f : X \to Y$ is a proper morphism, pushforward of cycles $f_{*} [V] = de g (V / f (V)) \cdot [\overline{f (V)}]$ descends to a homomorphism $f_{*} : A_{k} (X) \to A_{k} (Y)$ . (When $f (V)$ has strictly smaller dimension, $f_{*} [V] = 0$ .)

Flat pullback and Gysin maps. If $f : X \to Y$ is flat of relative dimension $d$ , flat pullback of cycles descends to $f^{*} : A_{k} (Y) \to A_{k + d} (X)$ . More generally, for a closed embedding $i : Y ↪ X$ of smooth varieties of pure codimension $d$ , the Gysin pullback (refined Gysin map) $i^{!} : A_{k} (X) \to A_{k - d} (Y)$ is defined by reduction to the normal cone; this is the central construction of modern intersection theory, developed below.

The Chow ring. On a smooth variety $X$ of dimension $n$ , the Chow groups $A^{*} (X) = ⨁_{i \geq 0} A^{i} (X)$ carry a graded-commutative ring structure. The multiplication

A^{i} (X) \times A^{j} (X) ⟶ A^{i + j} (X), (α, β) \mapsto α \cdot β

is the intersection product. Its construction proceeds by reduction to the diagonal: if $Δ : X ↪ X \times X$ is the diagonal embedding, the intersection product is defined by

α \cdot β = Δ^{!} (α \times β),

the Gysin pullback of the exterior product $α \times β \in A^{*} (X \times X)$ along the diagonal. The diagonal of a smooth variety is a regular embedding, which is what makes this pullback well-defined.

Degree map. For a complete (proper over $k$ ) variety $X$ , there is a degree map $de g : A_{0} (X) \to Z$ sending a 0-cycle $\sum n_{P} [P]$ to $\sum n_{P} [k (P) : k]$ . The intersection number of $n$ divisor classes on an $n$ -dimensional smooth projective variety is the degree of their iterated intersection product.

Relation to cohomology. For a smooth complex projective variety, there is a cycle class map $cl : A^{*} (X) \to H^{2 *} (X; Z)$ sending a codimension- $i$ cycle to its cohomology class in $H^{2 i} (X; Z)$ 04.03.01. This map is a ring homomorphism (it sends the intersection product to the cup product), but it is not an isomorphism in general: the kernel and cokernel of $cl$ measure the gap between algebraic and topological cycles, a central object of the theory of motives.

Key theorem with proof Intermediate+

Theorem (Chow ring of projective space; Bézout's theorem). Let $P_{k}^{n}$ be projective space over a field $k$ , and let $H \in A^{1} (P_{k}^{n})$ be the class of a hyperplane.

(i) (Chow ring of $P^{n}$ .) There is a ring isomorphism

A^{*} (P_{k}^{n}) ≅ Z [H] / (H^{n + 1}) .

In particular $A^{i} (P_{k}^{n}) = Z \cdot H^{i}$ for $0 \leq i \leq n$ , and $H^{n + 1} = 0$ .

(ii) (Bézout.) If $C, D \subset P_{k}^{2}$ are plane curves of degrees $m, n$ with no common irreducible component, then $C$ and $D$ meet in exactly $mn$ points counted with local intersection multiplicity.

Proof. Part (i). Every integral codimension- $i$ subvariety $V \subset P_{k}^{n}$ has a well-defined degree $de g (V)$ , namely the number of points in the intersection of $V$ with a general $(n - i)$ -plane. The degree map

de g : A^{i} (P_{k}^{n}) ⟶ Z, \sum n_{V} [V] \mapsto \sum n_{V} de g (V),

is well-defined on rational equivalence classes because degree is constant in flat $P^{1}$ -families. Any codimension- $i$ linear subspace $L^{i}$ has degree 1, and $H^{i} = [L^{i}]$ in $A^{i}$ . Since $de g (V) = de g (V) \cdot de g (L^{i})$ and $V$ is rationally equivalent to $de g (V) \cdot L^{i}$ (a standard deformation through the incidence correspondence ${(V, Λ) : V \cap Λ \neq = \emptyset}$ ), the degree map is an isomorphism $A^{i} \sim Z$ sending $H^{i} \mapsto 1$ for $0 \leq i \leq n$ .

The relation $H^{n + 1} = 0$ holds because $n + 1$ hyperplanes in general position in $P^{n}$ have empty intersection, and the class of the empty cycle is zero. It follows that the surjective map $Z [H] \to A^{*} (P_{k}^{n})$ sending $H \mapsto H$ descends to $Z [H] / (H^{n + 1}) \to A^{*} (P_{k}^{n})$ , which is a ring isomorphism by degree-counting in each codimension.

Part (ii) (Bézout). Let $C, D \subset P_{k}^{2}$ have degrees $m, n$ and no common component. By part (i), $[C] = m H$ and $[D] = n H$ in $A^{1} (P_{k}^{2})$ , since a plane curve of degree $d$ has class $d H$ . The intersection product is

[C] \cdot [D] = (m H) (n H) = mn H^{2} \in A^{2} (P_{k}^{2}),

and $H^{2}$ is the class of a point, so $[C] \cdot [D]$ is the 0-cycle " $mn$ points." The hypothesis that $C, D$ share no component means their intersection is proper (codimension $1 + 1 = 2$ , the expected codimension in $P^{2}$ ), so the scheme-theoretic intersection cycle $\sum_{P \in C \cap D} i_{P} (C, D) [P]$ — where $i_{P} (C, D) = dim_{k} (O_{P^{2}, P} / (f, g))$ is the local intersection multiplicity — equals the Chow-ring intersection product. Taking degrees,

P \in C \cap D \sum i_{P} (C, D) = de g (mn H^{2}) = mn . □

This theorem is the foundational pattern of all of enumerative geometry: compute an intersection number by computing a product in the Chow ring. The moving lemma of Chow ^[Chow1956] and the deformation-to-the-normal-conse construction of Fulton-MacPherson ^{[FultonMacPherson1978]} extend this pattern from the transverse setting of Bézout to arbitrary intersections on smooth varieties, including tangencies and excess intersections. The same pattern — degree equals product — gives the 27 lines on a smooth cubic surface (computed in $A^{*} (P^{2} blown up at six points)$ ) and the 3264 conics tangent to five given conics (computed in the Chow ring of the space of complete conics).

Bridge. The construction here builds toward 04.05.09 (Hodge index theorem) and 04.05.10 (Hirzebruch-Riemann-Roch), where the intersection product on surfaces and higher-dimensional varieties is sharpened into numerical invariants. The defining pattern appears again in 04.03.01 (sheaf cohomology) in a sharpened form: the cycle class map sends the Chow ring to the even cohomology ring, and the central insight is that intersection theory is the algebraic shadow of the cup product in cohomology. The intersection product generalises Chern-class computations on vector bundles 04.05.05, and the foundational reason intersection numbers are well-defined is exactly that rational equivalence absorbs the ambiguity of choosing representatives; this is precisely the statement that the Chow ring is a ring and not merely a graded abelian group. Putting these together, the bridge is that Bézout's theorem is the codimension-1 prototype of every enumerative computation, and the Chow ring is the universal setting where such computations live.

Exercises Intermediate+

Exercise 3 (medium, proof).

Compute the Chow ring of $X = P_{k}^{1} \times P_{k}^{1}$ , and use it to compute the self-intersection number of a fibre of each projection.

Hint

Use the projective bundle formula (or Künneth-type decomposition) for products of projective spaces.

Answer

Let $α, β \in A^{1} (X)$ be the classes of the fibres of the first and second projections, $[{p} \times P^{1}]$ and $[P^{1} \times {q}]$ . By the projective bundle formula (or direct rational-equivalence argument),

A^{*} (P^{1} \times P^{1}) ≅ Z [α, β] / (α^{2}, β^{2}) .

The class $α$ is pulled back from the first factor, where $H^{2} = 0$ in $A^{2} (P^{1}) = 0$ -cycles of degree 0; hence $α^{2} = 0$ ; similarly $β^{2} = 0$ . The mixed product is $α β = [pt]$ , the class of a point in $A^{2} (X) ≅ Z$ .

A fibre $F_{1} = [{p} \times P^{1}]$ has class $α$ . Its self-intersection is $α \cdot α = α^{2} = 0$ in $A^{2} (X)$ , so the self-intersection number of a fibre of the first projection is $0$ . Geometrically: two distinct fibres of $pr_{1}$ are disjoint, and the self-intersection captures this. Similarly the self-intersection of a fibre $F_{2}$ of $pr_{2}$ is $β^{2} = 0$ .

By contrast, the diagonal $Δ \subset P^{1} \times P^{1}$ has class $α + β$ (its degree is 1 in each factor), and its self-intersection is $(α + β)^{2} = 2 α β = 2$ , the Euler characteristic of $P^{1}$ — the Lefschetz fixed-point count.

Exercise 5 (medium, conceptual).

Why does the intersection product on $A^{*} (X)$ require $X$ to be smooth? What goes wrong on a singular variety?

Hint

The definition uses reduction to the diagonal, and relies on the diagonal being a regular embedding.

Answer

The intersection product on $A^{*} (X)$ is defined by $α \cdot β := Δ^{!} (α \times β)$ , the Gysin pullback of the exterior product along the diagonal $Δ : X ↪ X \times X$ . For this refined Gysin map $Δ^{!}$ to exist, the diagonal embedding must be a regular embedding (locally cut out by a regular sequence), which is equivalent to $X$ being smooth over the base.

On a singular variety, the diagonal is not a regular embedding: the local rings of $X \times X$ along $Δ$ are not regular sequences. As a result the naive intersection of two cycles is not well-defined modulo rational equivalence — the choice of representative cycles affects the rational-equivalence class of the intersection.

For singular varieties one therefore keeps only a partial structure: the operational Chow ring $A^{*} (X)$ of Fulton, whose elements are operations $A_{*} (X) \to A_{*- i} (X)$ rather than cycle classes. This operational ring agrees with the cycle-level Chow ring when $X$ is smooth, and retains enough structure to define Chern classes and intersection numbers, but it is weaker than a true cycle-level intersection product.

Exercise 6 (hard, proof).

State the excess intersection formula for a regular embedding and prove its special case: the self-intersection formula (see Full proof set below for the full argument). Concretely, for a closed embedding $i : Y ↪ X$ of smooth varieties of codimension $d$ with normal bundle $N_{Y / X}$ , show that for every $α \in A_{*} (Y)$ ,

i^{!} i_{*} α = c_{d} (N_{Y / X}) \cap α .

Hint

Reduce to the case where $Y$ is the zero-section of its normal bundle; there $i^{!}$ is division by the top Chern class of the normal bundle.

Answer

The excess intersection formula concerns the following situation: a fibre square

Y^{'} ↓ i^{'} X^{'} g f Y ↓ i X

with $i : Y ↪ X$ a regular embedding of codimension $d$ , $Y^{'}, X^{'}$ smooth (or more generally $i^{'}$ a regular embedding of codimension $d - e$ , with $e$ the excess), and normal bundles $N = N_{Y / X}$ , $N^{'} = N_{Y^{'} / X^{'}}$ . The excess bundle is $E = g^{*} N / N^{'}$ , a rank- $e$ bundle on $Y^{'}$ . The excess intersection formula states

f^{*} i_{*} α = i_{*}^{'} (c_{e} (E) \cap g^{*} α) .

For the self-intersection formula, take $X^{'} = Y$ , $f = i : Y ↪ X$ , $Y^{'} = Y$ , $i^{'} = id$ . Then $N^{'} = N_{Y / Y} = 0$ , $E = i^{*} N_{Y / X} = N_{Y / X}$ (rank $d$ , so $e = d$ ), and the formula reads

i^{*} i_{*} α = id_{*} (c_{d} (N_{Y / X}) \cap α) = c_{d} (N_{Y / X}) \cap α,

with $i^{!}$ replacing $i^{*}$ for the refined version. The proof that $i^{!} i_{*} α = c_{d} (N_{Y / X}) \cap α$ proceeds by deformation to the normal cone (see the Full proof set), reducing to the model case where $X =$ total space of $N_{Y / X}$ and $Y$ is the zero-section, where $i^{!} i_{*}$ is computed by the Thom isomorphism and division by the top Chern class. $□$

Exercise 7 (hard, conceptual).

Outline how the Chow ring computes the number of lines on a smooth cubic surface $S \subset P^{3}$ .

Hint

$S$ is the blow-up of $P^{2}$ at six points in general position; use the Chow ring of the blow-up.

Answer

A smooth cubic surface $S \subset P^{3}$ is isomorphic to the blow-up of $P^{2}$ at six points $p_{1}, \dots, p_{6}$ in general position (no three collinear, no six on a conic). Let $π : S \to P^{2}$ be the blow-down, $H = π^{*} ([line])$ the pullback of the hyperplane class, and $E_{1}, \dots, E_{6}$ the classes of the six exceptional curves. The Chow ring is

A^{*} (S) ≅ Z [H, E_{1}, \dots, E_{6}] / (H^{2} = 1, E_{i} E_{j} = - δ_{ij}, H E_{i} = 0) .

A line on $S$ is a curve $L$ with $L^{2} = - 1$ and $L \cdot H = 1$ (self-intersection $- 1$ and degree 1 as a curve in $P^{3}$ ). Write $L \sim d H - \sum m_{i} E_{i}$ . The conditions $L \cdot H = 1$ and $L^{2} = - 1$ give

d = 1, d^{2} - \sum m_{i}^{2} = 1 - \sum m_{i}^{2} = - 1,

so $\sum m_{i}^{2} = 2$ . With $L$ effective ( $m_{i} \geq 0$ ) and $K_{S} = - 3 H + \sum E_{i}$ , adjunction $L \cdot (L + K_{S}) = - 2$ forces $L \cdot K_{S} = - 1$ , giving $- 3 + \sum m_{i} = - 1$ , hence $\sum m_{i} = 2$ .

The only integer solutions to $\sum m_{i} = 2$ and $\sum m_{i}^{2} = 2$ with $m_{i} \in {0, 1}$ are the $(2 6) = 15$ pairs ${i, j}$ , giving the 15 classes $H - E_{i} - E_{j}$ . Together with the 6 exceptional classes $E_{i}$ and the 6 "anti-exceptional" classes $2 H - \sum_{i \neq = j} E_{i}$ (one per $j$ ), the total is $15 + 6 + 6 = 27$ . So $S$ carries exactly 27 lines.

This computation — carried out entirely in the Chow ring of the blow-up — is the classical result of Cayley and Salmon (1849), recovered as a clean enumerative calculation. The same Chow-ring machinery, applied to the space of complete conics, gives the 3264 conics tangent to five given conics ^{[EisenbudHarris2016]}.

Lean formalization Intermediate+

lean_status: none — Mathlib does not currently contain a global theory of Chow groups, algebraic cycles modulo rational equivalence, or the intersection product. The gap is substantial: see the Mathlib gap analysis frontmatter note for details. Until Mathlib develops cycle-level intersection theory, the constructions and enumerative theorems of this unit (Bézout's count, the 27 lines on a cubic surface, the excess intersection formula) cannot be stated in Lean.

Advanced results Master

Deformation to the normal cone. The single most important technical device of modern intersection theory, due to Fulton and MacPherson ^{[FultonMacPherson1978]}. Given a closed embedding $i : Y ↪ X$ , one constructs a flat family over $A^{1}$ whose general fibre is $X$ and whose special fibre is the normal cone $C_{Y} X$ — a cone over $Y$ containing the normal bundle $N_{Y / X}$ as an open dense subcone when $Y$ is a regular embedding. The family is built as $M_{Y}^{\circ} X := Bl_{Y \times {\infty}} (X \times P^{1}) ∖ Bl_{Y \times {\infty}} (X \times {\infty})$ , the deformation space. Because intersection numbers are constant in flat families, intersection questions about $Y ↪ X$ reduce to intersection questions about the zero-section of the normal cone, which are tractable.

The refined Gysin map. For a regular embedding $i : Y ↪ X$ of codimension $d$ , deformation to the normal cone defines a refined Gysin map $i^{!} : A_{k} (X) \to A_{k - d} (Y)$ for any $k$ -cycle on $X$ , even when its support does not meet $Y$ transversely. The construction: given a cycle $α$ on $X$ , deform $X$ to $C_{Y} X$ , take the inverse-image cycle on the normal cone, and cap with the Segre class of $C_{Y} X / N_{Y / X}$ . This is the foundation of the intersection product (via reduction to the diagonal) and of the excess intersection formula.

Excess intersection formula. For a fibre square of regular embeddings with excess bundle $E$ of rank $e$ ,

f^{*} i_{*} α = i_{*}^{'} (c_{e} (E) \cap g^{*} α),

the excess intersection formula. It corrects the naive pullback by the top Chern class of the excess bundle, and it is the rigorous statement that replaces the false "intersection number = product of degrees" when intersections are not transverse. The self-intersection formula $i^{!} i_{*} α = c_{d} (N_{Y / X}) \cap α$ is the special case $e = d$ .

Chern classes as operators. For a vector bundle $E$ of rank $r$ on an $n$ -dimensional smooth variety $X$ , the Chern classes $c_{i} (E) \in A^{i} (X)$ act by capping: $c_{i} (E) \cap : A_{k} (X) \to A_{k - i} (X)$ . The top Chern class $c_{r} (E) \cap [X]$ is the class of the zero-locus of a general section of $E$ . This operational definition extends to singular varieties and gives Chern classes for all vector bundles without smoothness, a major advantage of Fulton's approach over the classical one via projective bundles.

Segre classes and the Segre-Chern relation. The Segre class $s (E) = c (E)^{- 1}$ satisfies $s (E) c (E) = 1$ in $A^{*} (X)$ . Segre classes compute the contribution of the excess cone in the refined Gysin map; together with the Chern class they form the basic toolkit of modern enumerative geometry.

Grothendieck-Riemann-Roch. For a proper morphism $f : X \to Y$ of smooth varieties and a coherent sheaf $F$ on $X$ ,

ch (f_{!} [F]) \cdot td (T_{Y}) = f_{*} (ch (F) \cdot td (T_{X})) in A^{*} (Y) \otimes Q,

where $ch$ is the Chern character, $td$ the Todd class, and $f_{!}$ the $K$ -theoretic pushforward. GRR identifies intersection-theoretic and $K$ -theoretic pushforwards, and specialises to Hirzebruch-Riemann-Roch 04.05.10 when $Y = pt$ , recovering $χ (X, F) = de g (ch (F) \cdot td (T_{X}))$ .

Enumerative applications. Three classical counts sit at the heart of enumerative geometry, all computed in Chow rings: (i) Bézout's theorem (this unit, Key theorem), the count of $mn$ intersection points of plane curves of degrees $m, n$ ; (ii) the 27 lines on a smooth cubic surface (Exercise 7), computed in $A^{*}$ of the blow-up of $P^{2}$ at six points; (iii) the 3264 conics tangent to five given conics ^{[EisenbudHarris2016]}, computed by Chasles in 1864 and recovered by Fulton-MacPherson via the Chow ring of the space of complete conics.

Relation to $K$ -theory and motives. The cycle class map $A^{*} (X) \to H^{2 *} (X; Z)$ factors through algebraic $K$ -theory via the Chern character $ch : K_{0} (X) \otimes Q \to A^{*} (X) \otimes Q$ (an isomorphism for smooth $X$ , by Grothendieck-Riemann-Roch). The motivic viewpoint (Voevodsky) promotes Chow groups to a universal cohomology theory: Chow groups are the morphism groups in Voevodsky's category of effective motives, and the standard conjectures of Grothendieck govern when the cycle class map becomes an isomorphism.

Synthesis. The Chow ring $A^{*} (X)$ of a smooth variety $X$ is the algebraic counterpart of the cohomology ring, with the intersection product playing the role of the cup product, and the foundational reason this works is that rational equivalence absorbs the ambiguity in choosing cycle representatives so that intersections are well-defined at the level of classes; this is exactly the content of the moving lemma and the deformation-to-the-normal-conse construction. The refined Gysin map generalises the intersection product to non-transverse situations through the excess intersection formula, the central insight being that an "over-determined" intersection is corrected by the top Chern class of the excess bundle. Putting these together, the bridge is that every classical enumerative count — from Bézout's $mn$ to the 27 lines and the 3264 conics — becomes a single multiplication in an appropriate Chow ring, and the cycle class map to singular cohomology 04.03.01 shows that this purely algebraic computation faithfully records topological intersection data. Operational Chow groups extend the theory to singular varieties, and Grothendieck-Riemann-Roch identifies the Chow ring with rationalised $K$ -theory, so the central insight is that intersection theory is the universal algebraic framework in which enumerative geometry, characteristic classes, and Riemann-Roch formulae all live.

Full proof set Master

Proposition (Self-intersection formula via deformation to the normal cone). Let $X$ be a smooth variety and $i : Y ↪ X$ a closed embedding of smooth varieties of codimension $d$ with normal bundle $N = N_{Y / X}$ . For every $α \in A_{*} (Y)$ , the refined Gysin pullback satisfies

i^{!} i_{*} α = c_{d} (N) \cap α in A_{*- d} (Y) .

Proof. We use the deformation to the normal cone. Let $M_{Y}^{\circ} X$ be the deformation space: the complement of the proper transform of $X \times {\infty}$ in the blow-up $Bl_{Y \times {\infty}} (X \times P^{1})$ . This is a smooth variety (since $X$ and $Y$ are smooth and the centre $Y \times {\infty}$ is regularly embedded) carrying a flat morphism $ρ : M_{Y}^{\circ} X \to P^{1}$ , with general fibre $ρ^{- 1} (t) ≅ X$ for $t \neq = \infty$ and special fibre

ρ^{- 1} (\infty) = C_{Y} X \supseteq N_{Y / X}

equal to the normal cone of $Y$ in $X$ , containing the normal bundle $N_{Y / X}$ as an open dense subcone. Because $Y$ is regularly embedded in $X$ (both smooth), the normal cone is the normal bundle: $C_{Y} X = N_{Y / X}$ , and the special fibre of $ρ$ is the total space of $N_{Y / X}$ .

The cycle $i_{*} α \in A_{*} (X)$ is represented by a cycle $α$ on $M_{Y}^{\circ} X$ , flat over $P^{1}$ , whose restriction to the general fibre is $i_{*} α$ and whose restriction to the special fibre is a cycle $α_{\infty}$ on the total space of $N$ . By principle of continuity (intersection numbers are constant in flat families), the refined Gysin map $i^{!} i_{*} α$ equals the Gysin map of the zero-section $ι : Y ↪ N$ applied to $α_{\infty}$ .

It remains to compute $ι^{!} ι_{*} α$ for the zero-section of a vector bundle $N$ of rank $d$ on $Y$ . By the Thom isomorphism for Chow groups (a consequence of the homotopy property and the projective bundle formula),

p^{*} : A_{*} (Y) \sim A_{*+ d} (N)

is an isomorphism, where $p : N \to Y$ is the projection. The class $ι_{*} α \in A_{*} (N)$ corresponds under $p^{*}$ to a unique class on $Y$ ; a computation in the projective bundle $P (N \oplus O)$ shows that $ι^{*} ι_{*} α = c_{d} (N) \cap α$ (the excess intersection formula for the zero-section, where the excess bundle is $N$ itself).

Combining: the deformation reduces the general self-intersection $i^{!} i_{*}$ to the zero-section case, and the zero-section case is computed by the Thom isomorphism. Hence $i^{!} i_{*} α = c_{d} (N_{Y / X}) \cap α$ in $A_{*- d} (Y)$ . $□$

Corollary (excess intersection formula). In the fibre square of regular embeddings with excess bundle $E$ of rank $e$ , $f^{*} i_{*} α = i_{*}^{'} (c_{e} (E) \cap g^{*} α)$ . The proof is the same deformation argument, with the normal cone $C_{Y^{'}} X^{'}$ replacing $N_{Y^{'} / X^{'}}$ and the Segre class $s (E)$ correcting the difference; the top Chern class $c_{e} (E)$ emerges from $s (E) c (E) = 1$ .

Connections Master

Sheaf cohomology 04.03.01 — the cycle class map $A^{*} (X) \to H^{2 *} (X; Z)$ sends the intersection product to the cup product in cohomology; for smooth complex projective varieties this is a ring homomorphism whose kernel and cokernel measure the gap between algebraic and topological cycles.
Weil divisor 04.05.01 — the first Chow group $A^{1} (X)$ is the divisor class group $Cl (X)$ on a smooth variety; intersection theory generalises the divisor-class picture to cycles of arbitrary codimension, with divisors as the codimension-one layer.
Hodge index theorem 04.05.09 — the intersection form on a smooth projective surface has signature $(1, ρ - 1)$ on $NS (X) \otimes R$ ; this numerical consequence of the intersection product governs the geometry of surfaces and is the surface-level input to the minimal model program.
Hirzebruch-Riemann-Roch 04.05.10 — the formula $χ (X, F) = de g (ch (F) \cdot td (T_{X}))$ evaluates Euler characteristics by computing intersection numbers in the Chow ring; HRR is the dimension-0 face of Grothendieck-Riemann-Roch.
Ample line bundle 04.05.05 — the Nakai-Moishezon criterion for ampleness is phrased in terms of intersection numbers $(L^{r} \cdot Y) > 0$ , making ampleness a positivity condition on intersection-theoretic data.
Projective space 04.07.01 — the Chow ring $A^{*} (P^{n}) = Z [H] / (H^{n + 1})$ is the foundational computation from which Bézout's theorem and every projective enumerative count follow.
Birational geometry and the minimal model program 04.17.01 — intersection numbers $(D \cdot C)$ of divisors against curves define the nef and ample cones, and the MMP is the classification of varieties by the position of the canonical class in this cone of intersection numbers.

Historical & philosophical context Master

The rigorous theory of intersection numbers on algebraic varieties is one of the harder-won achievements of 20th-century geometry, and it solved a problem the Italian school had handled brilliantly but without secure foundations. The classical geometers — Guido Castelnuovo, Federigo Enriques, Francesco Severi — computed intersection numbers of curves on surfaces fluently in the years around 1900, but their arguments depended on "moving" cycles through families whose existence was taken on geometric faith. The school collapsed in a series of counterexamples (Severi's claimed proof of the Hodge conjecture for divisors on surfaces, later shown to rely on unjustified moving arguments), and the field awaited a rigorous foundation.

The first rigorous foundation came from Wei-Liang Chow, who in "On the multiplication of algebraic cycles" ^[Chow1956] introduced rational equivalence of algebraic cycles and proved the moving lemma: every cycle can be moved into general position within its rational-equivalence class, so the intersection product of rational-equivalence classes is well-defined. Chow's construction gave the ring $A^{*} (X)$ of algebraic cycles modulo rational equivalence, now called the Chow ring. Chow's 1956 paper, together with earlier work of W. Hodge and D. Pedoe [Methods of Algebraic Geometry, 1947–1954], placed the Italian-school intersection numbers on a secure algebraic footing.

The second revolution was Alexander Grothendieck's reformulation of the theory in the language of schemes, Chern classes, and $K$ -theory, in the Éléments de Géométrie Algébrique (EGA) and the Séminaire de Géométrie Algébrique (SGA 6, 1971). Grothendieck recast the Chow ring as the target of a universal Chern character from $K_{0} (X)$ , and conjectured the Riemann-Roch theorem in its full generality — proved by Armand Borel and Jean-Pierre Serre in 1958 and extended by Grothendieck to the Grothendieck-Riemann-Roch theorem, identifying $K$ -theoretic and Chow-theoretic pushforwards up to the Todd class.

The third revolution, which produced the modern theory in its current form, was the work of William Fulton and Robert MacPherson in the late 1970s ^{[FultonMacPherson1978]}. Their two innovations — deformation to the normal cone and the excess intersection formula — gave a refined Gysin map $i^{!}$ for any closed embedding, not only regular ones, and made intersection theory work in non-transverse situations and on singular varieties (via operational Chow groups). Fulton's book Intersection Theory ^[Fulton1984] (1984, 2nd edition 1998) became the canonical reference, systematising the theory as an oriented Borel-Moore homology theory satisfying a short list of axioms.

The enumerative payoffs were immediate. The classical count of 3264 conics tangent to five given conics — first computed by Michel Chasles in 1864 and refined by Hieronymus Zeuthen — was re-derived in modern language by Fulton and MacPherson via the Chow ring of the space of complete conics. Eisenbud and Harris's 3264 and All That ^{[EisenbudHarris2016]} (2016) made the modern theory accessible to a generation of geometers, with the eponymous count as its capstone.

Philosophically, intersection theory is the algebraic counterpart of algebraic topology's cup product, and the cycle class map $A^{*} (X) \to H^{2 *} (X; Z)$ is the bridge. The failure of this map to be an isomorphism — the existence of cohomology classes not represented by algebraic cycles — is the content of the Hodge conjecture and the standard conjectures of Grothendieck, which remain among the deepest open problems in mathematics. From this viewpoint intersection theory is not a computational tool but a window onto the relationship between algebra and topology: the Chow ring is the universal algebraic cohomology theory, and its comparison with singular, étale, and crystalline cohomology is the heart of modern algebraic geometry.

Bibliography Master

@book{Fulton1998,
  author    = {William Fulton},
  title     = {Intersection Theory},
  edition   = {2nd},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {2},
  publisher = {Springer-Verlag},
  address   = {Berlin},
  year      = {1998}
}

@book{EisenbudHarris2016,
  author    = {David Eisenbud and Joe Harris},
  title     = {3264 and All That: A Second Course in Algebraic Geometry},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {2016}
}

@book{Hartshorne1977,
  author    = {Robin Hartshorne},
  title     = {Algebraic Geometry},
  series    = {Graduate Texts in Mathematics},
  volume    = {52},
  publisher = {Springer-Verlag},
  address   = {New York},
  year      = {1977},
  note      = {Appendix A: Intersection Theory}
}

@incollection{Chow1956,
  author    = {Wei-Liang Chow},
  title     = {On the multiplication of algebraic cycles},
  booktitle = {Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz},
  publisher = {Princeton University Press},
  address   = {Princeton},
  year      = {1957}
}

@unpublished{FultonMacPherson1978,
  author    = {William Fulton and Robert MacPherson},
  title     = {Intersection Theory},
  note      = {Erfurt preprint, 1978; published in expanded form as \cite{Fulton1998}},
  year      = {1978}
}

@book{HodgePedoe1952,
  author    = {W. V. D. Hodge and D. Pedoe},
  title     = {Methods of Algebraic Geometry},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {1947--1952}
}

@book{Vakil2017,
  author    = {Ravi Vakil},
  title     = {The Rising Sea: Foundations of Algebraic Geometry},
  note      = {Preprint, \S\,20--22 on intersection theory and Chow groups},
  year      = {2017}
}

@phdthesis{Voisin2003,
  author    = {Claire Voisin},
  title     = {Hodge Theory and Complex Algebraic Geometry, {I}},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {2002},
  note      = {Cycle class map, standard conjectures}
}

Prerequisites

04.03.01
04.05.01

Tier anchors

beginner: Eisenbud-Harris 2016 3264 and All That (Cambridge UP) Ch. 1 — counting how subvarieties meet, with multiplicity, in plain language.
intermediate: Eisenbud-Harris 2016 3264 and All That Ch. 1–2; Hartshorne 1977 Algebraic Geometry (Springer) App. A — Chow groups, the Chow ring, and the intersection product.
master: Fulton 1984/1998 Intersection Theory (Springer); Eisenbud-Harris 2016 — full proofs of the intersection product via reduction to the diagonal, deformation to the normal cone, and the excess intersection formula.

References

Fulton — Intersection Theory · Springer, 2nd ed. 1998 (1st ed. 1984). The canonical reference: algebraic cycles, rational equivalence, Chow groups $A_{*} (X)$ , the intersection product on smooth varieties via reduction to the diagonal, Chern classes as operators on Chow groups, deformation to the normal cone, and the excess intersection formula. Chapter 1–2 build the theory from axioms; Chapter 6 proves the self-intersection formula and the excess intersection formula; Chapters 15–18 give enumerative applications.
Eisenbud-Harris — 3264 and All That · Cambridge UP, 2016. A graduate introduction built around enumerative problems: the Chow ring of projective space $A^{*} (P^{n}) = Z [H] / (H^{n + 1})$ , the Chow ring of a blow-up, the 27 lines on a smooth cubic surface, and the capstone computation of the number 3264 of conics tangent to five given conics in $P^{2}$ , carried out in the Chow ring of the space of complete conics.
Hartshorne — Algebraic Geometry · Springer GTM 52, 1977. Appendix A (Intersection Theory) defines Chern classes on a nonsingular variety via the Chow ring $A^{*} (X)$ , states the intersection product through reduction to the diagonal, and sketches Grothendieck-Riemann-Roch. §I.7 covers Bézout's theorem and the degree of a projective variety.
Chow — On the multiplication of algebraic cycles · Wei-Liang Chow, Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz, Princeton UP, 1957 (the multiplication of cycles paper circulates as Chow 1956). Defines rational equivalence of algebraic cycles and proves the moving lemma that endows rational-equivalence classes with a well-defined intersection product, giving the Chow ring $A^{*} (X)$ .
Fulton-MacPherson — Intersection Theory (Erfurt notes) · Fulton and MacPherson, 1978 Erfurt preprint / published as Fulton 1984 Ch. 6–9. The deformation-to-the-normal-conse construction and the excess intersection formula, defining a refined Gysin map $i^{!}$ for any closed embedding, the technical core of modern intersection theory on singular and non-transverse situations.

Estimated time

beginner: 15m
intermediate: 45m
master: 90m