Intersection Theory — Basics

Intersection theory assigns algebraic invariants to the geometric act of two subvarieties meeting inside an ambient variety. One-line statement: for subvarieties $V$ , $W$ of a smooth variety $X$ meeting "properly," there is a well-defined intersection cycle $V \cdot W$ , a sum of components of $V \cap W$ weighted by intersection multiplicities, living in the Chow ring $A^{*} (X)$ . Key result: the Chow ring exists and the intersection product is commutative, associative, and compatible with proper pushforward and flat pullback; over $P^{n}$ it specializes to Bézout's theorem. Where it applies: enumerative geometry, algebraic cycles, characteristic classes, moduli theory, and Gromov–Witten invariants. Who/when: classical roots in Poncelet, Bézout, and Schubert (19th c.); rigorous foundations by van der Waerden, Weil, Severi, Chevalley, and definitively Fulton–MacPherson (1970s–80s).

Intersection theory is the branch of algebraic geometry that makes precise the statement "two curves of degrees $d$ and $e$ in the plane meet in $d e$ points." Given a smooth variety $X$ and two subvarieties $V$ and $W$ , one wants a cycle class $[V] \cdot [W]$ recording how they intersect, counted with multiplicities and supported on $V \cap W$ . The natural home for these classes is the Chow group $A_{k} (X)$ of $k$ -dimensional cycles modulo rational equivalence; when $X$ is smooth these assemble into a graded Chow ring $A^{*} (X)$ under intersection product. The surprising fact is that even when $V$ and $W$ meet badly — in excess dimension, or non-transversally, or with embedded components — there is still a canonical intersection class, produced by Fulton and MacPherson via deformation to the normal cone, with no recourse to moving the cycles. This robustness is what makes intersection theory a computational engine rather than a mere heuristic.

The Core Insight

The number of points in which two subvarieties meet is not an invariant of the subvarieties — but their rational equivalence classes are, and intersection respects rational equivalence. So the right move is: (1) replace cycles by their classes in the Chow group, (2) define a product on these classes, and (3) prove the product is independent of all choices. The two technologies that make this work are moving lemmas (deform cycles into general position so naive multiplicities apply) and, more powerfully, the deformation to the normal cone (which produces the intersection class intrinsically, even with no genericity at all).

Formal Statement

Cycles and rational equivalence

Let $X$ be an algebraic scheme (separated, finite type) over a field $k$ . A $k$ -cycle on $X$ is a finite formal $Z$ -linear combination $\sum n_{i} [V_{i}]$ of $k$ -dimensional integral closed subvarieties $V_{i} \subseteq X$ . The group of all such is $Z_{k} (X)$ .

For a $(k + 1)$ -dimensional subvariety $W \subseteq X$ and a nonzero rational function $r \in k (W)^{\times}$ , the principal divisor is

div (r) = V \sum ord_{V} (r) [V] \in Z_{k} (X),

⟳the sum runs over codimension-one subvarieties

V

W

;

ord_{V} (r)

is the length-based order of vanishing/pole,

ord_{V} (r) = length_{O_{W, V}} (O_{W, V} / (r))

when

r

is a non-zero-divisor, extended additively to ratios.

Two cycles are rationally equivalent if their difference is a $Z$ -combination of principal divisors. The Chow group is

A_{k} (X) = Z_{k} (X) / {rational equivalences}, A_{*} (X) = k ⨁ A_{k} (X) .

⟳quotient the free abelian group on subvarieties by the subgroup generated by all

div (r)

over all

W

and

r

; this is the algebraic analogue of homology with principal divisors playing the role of boundaries.

Intersection multiplicity, the proper case

Let $X$ be a smooth $n$ -dimensional variety, and let $V$ , $W$ be subvarieties of dimensions $dim V$ , $dim W$ meeting properly, i.e. every component $Z$ of $V \cap W$ has the expected dimension $dim V + dim W - n$ . The intersection multiplicity of $V$ and $W$ along $Z$ is Serre's $Tor$ -formula:

i (Z; V, W) = j \geq 0 \sum (- 1)^{j} length_{O_{X, Z}} (Tor_{j}^{O_{X, Z}} (O_{V, Z}, O_{W, Z})) .

⟳the naive length

length (O_{V, Z} \otimes O_{W, Z})

overcounts when the intersection is non-Cohen–Macaulay; the alternating sum of higher Tors corrects it, and Serre proved this is finite, nonnegative, and gives the geometrically correct multiplicity.

The intersection product is then

V \cdot W = Z \subseteq V \cap W \sum i (Z; V, W) [Z] \in A_{d i m V + d i m W - n} (X) .

⟳sum over the components

Z

V \cap W

(all of expected dimension by properness), weighting each by its Serre multiplicity; well-definedness on classes requires the moving lemma or deformation to the normal cone.

Chow ring structure (smooth case)

If $X$ is a smooth quasi-projective variety over a field, then $A^{*} (X) := ⨁_{k} A^{k} (X)$ — indexed by codimension $A^{k} (X) = A_{n - k} (X)$ — carries a graded-commutative ring structure with product extending [4], satisfying:

Commutativity: $α \cdot β = β \cdot α$ .
Associativity: $(α \cdot β) \cdot γ = α \cdot (β \cdot γ)$ .
Projection formula: for proper $f : X \to Y$ , $f_{*} (α \cdot f^{*} β) = f_{*} α \cdot β$ .
Pullback: for any morphism $f : X^{'} \to X$ of smooth varieties there is a ring homomorphism $f^{*} : A^{*} (X) \to A^{*} (X^{'})$ .
The class $[X] \in A^{0} (X)$ is the multiplicative identity.

Bézout's theorem

Let $V_{1}, \dots, V_{r}$ be hypersurfaces in $P^{n}$ of degrees $d_{1}, \dots, d_{r}$ meeting properly. Then, counted with intersection multiplicities,

de g (V_{1} \cdot V_{2} \dots V_{r}) = d_{1} d_{2} \dots d_{r} .

⟳

A^{*} (P^{n}) ≅ Z [h] / (h^{n + 1})

with

h = [hyperplane]

; a degree-

d

hypersurface has class

d h

, so the product is

(d_{1} h) \dots (d_{r} h) = (\prod d_{i}) h^{r}

, and

de g (h^{n}) = 1

gives the point count.

Why It Holds / Derivation

The intersection product via deformation to the normal cone

We sketch the Fulton–MacPherson construction of $V \cdot W$ for $V, W \subseteq X$ with $X$ smooth, requiring no genericity.

Step 1 — Reduce to intersecting with the diagonal. The intersection

V \cap W

inside

X

equals

(V \times W) \cap Δ_{X}

inside

X \times X

, where

Δ_{X} ≅ X

is the diagonal. Since

X

is smooth,

Δ_{X} ↪ X \times X

is a regular embedding (locally cut out by a regular sequence). This converts the general problem into intersecting a cycle with a regularly embedded smooth subvariety. Step 2 — Deform to the normal cone. For a closed embedding

i : Z ↪ Y

(here

Z = Δ_{X}

Y = X \times X

), construct the family

M_{Z}^{\circ} Y = Bl_{Z \times {0}} (Y \times P^{1}) ∖ Bl_{Z \times {0}} (Y \times {0}),

⟳blow up

Y \times P^{1}

along

Z \times {\infty}

(or

{0}

); the fiber over generic

t

Y

, but the fiber over the special point degenerates to the normal cone

C_{Z} Y

, geometrically "zooming in" on

Z

This gives a flat family over $P^{1}$ whose general fiber is $Y$ and whose special fiber is the normal cone $C_{Z} Y$ . When $i$ is a regular embedding, $C_{Z} Y = N_{Z} Y$ is the normal bundle.

Step 3 — Specialize the cycle. A cycle

α

Y

(here

α = [V \times W]

) deforms flatly across the family; its limit in the special fiber is a cycle on the normal cone, denoted

σ (α) \in Z_{*} (C_{Z} Y)

. Step 4 — Intersect with the zero section via Gysin. The normal bundle

N = N_{Z} Y

is a vector bundle over

Z

of rank

d = codim

. There is a Gysin/refined-intersection map (inverse to flat pullback on Chow groups of bundles)

s_{N}^{*} : A_{k} (N) \sim A_{k - d} (Z),

⟳

A_{*} (N) ≅ A_{*} (Z)

via flat pullback

π^{*}

(homotopy invariance of Chow groups for affine bundles); the Gysin map is the inverse composed with the degree shift, formally "intersecting with the zero section."

Step 5 — Assemble. The intersection class is

V \cdot W := (pr_{V})_{*} (s_{N}^{*} σ ([V \times W])) \in A_{d i m V + d i m W - n} (V \cap W),

⟳read backwards — degenerate

V \times W

into the normal cone of the diagonal, intersect with the zero section using the Gysin isomorphism, and push the result to

V \cap W

; one then checks this refines to a class in

A_{*} (X)

matching [4] in the proper case.

Step 6 — Verify Serre's formula in the proper case. When the intersection is proper,

σ ([V \times W])

is supported on the right-dimensional pieces and

s_{N}^{*}

extracts exactly the multiplicities

i (Z; V, W)

of [3]; the equivalence of the cone-theoretic and Tor-theoretic multiplicities is a theorem of Fulton (and, for the local algebra, Serre).

■

William Fulton & Robert MacPherson | 1970s–1980s

Fulton (b. 1939) and MacPherson (b. 1944) recast intersection theory on a fully rigorous, characteristic-free, deformation-theoretic footing. Their 1981 memoir Categorical Framework for the Study of Singular Spaces and Fulton's 1984 monograph Intersection Theory replaced the older moving-lemma and Tor-formula approaches with deformation to the normal cone, dispensing with smoothness of the ambient space where possible and producing refined classes supported on the actual intersection. This made bivariant theory, excess-intersection formulas, and Riemann–Roch without denominators into systematic tools. Earlier rigor came from B. L. van der Waerden, André Weil (Foundations of Algebraic Geometry, 1946), Wei-Liang Chow, and Jean-Pierre Serre, who supplied the Tor multiplicity. Fulton's book remains the field's standard reference.

Building Intuition

Think of intersection theory as a bookkeeping system that survives all the ways geometry can go wrong. Two lines in the plane meet in one point — unless they are parallel, in which case they meet "at infinity" ( $P^{2}$ fixes this), or unless they coincide, in which case they meet in a whole line (excess dimension). Bézout's promise of $d e$ points is kept only after three corrections: work projectively (no points lost to infinity), work over $\overline{k}$ (no points lost to non-existence of roots), and count with multiplicity (a tangency counts twice, a triple contact thrice).

The deep idea is that you should not measure intersection by looking at $V \cap W$ as a set, but by recording how $V$ approaches $W$ to first order. That first-order data is the normal cone — the algebraic shadow of "the directions from which $V$ comes in." When $V$ and $W$ meet transversally, the cone is a plain bundle and you just count points. When they are tangent, the cone bends, and the bending is the excess multiplicity. Deformation to the normal cone is a slow-motion film of the collision; intersection multiplicity is what you read off the final frame.

Two conics meeting: transversal vs. tangential intersection, illustrating multiplicity

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Worked Examples

Bézout in the plane: a cubic meets a conic

Given: In

P^{2}

over

C

, let

C

be a smooth cubic (

d_{1} = 3

) and

Q

a smooth conic (

d_{2} = 2

), meeting properly. Find: the number of intersection points counted with multiplicity. Steps.

The Chow ring is $A^{*} (P^{2}) = Z [h] / (h^{3})$ , where $h = [line]$ satisfies $de g (h^{2}) = 1$ .
Class of $C$ : a degree-3 curve has $[C] = 3 h$ . Class of $Q$ : $[Q] = 2 h$ .
Intersection product: $[C] \cdot [Q] = (3 h) (2 h) = 6 h^{2}$ .
Degree: $de g (6 h^{2}) = 6 \cdot de g (h^{2}) = 6$ .

Answer:

6 points (with multiplicity)

. If, say, two coincide via a simple tangency, one sees 5 distinct points with multiplicities

2, 1, 1, 1, 1

Self-intersection of a curve on a surface

Given: On the surface

X = P^{1} \times P^{1}

, let

f = {p} \times P^{1}

and

g = P^{1} \times {q}

be the two ruling fibers. The Chow ring is

A^{*} (X) = Z [f, g] / (f^{2}, g^{2})

with

de g (f \cdot g) = 1

. Find: (a)

f \cdot g

, (b) the self-intersection

f^{2}

, (c) the self-intersection of the diagonal-type class

D = f + g

. Steps.

(a) $f$ and $g$ meet transversally in one point: $f \cdot g = 1$ point, $de g = 1$ .
(b) Two distinct horizontal fibers ${p} \times P^{1}$ and ${p^{'}} \times P^{1}$ are disjoint and rationally equivalent to $f$ ; hence $f^{2} = 0$ . (The relation $f^{2} = 0$ encodes "a fiber moves off itself.")
(c) $D^{2} = (f + g)^{2} = f^{2} + 2 f \cdot g + g^{2} = 0 + 2 (1) + 0 = 2$ .

Answer:

f \cdot g = 1, f^{2} = 0, D^{2} = 2

Common Misconceptions

"Intersection multiplicity is just $dim_{k} O_{V, Z} \otimes O_{W, Z}$ ." This naive length is correct only when the local intersection is Cohen–Macaulay (e.g. transversal, or hypersurface-against-hypersurface). In general it overcounts, and you must use the full alternating Tor-sum [3]. Serre's celebrated nonnegativity, vanishing, and positivity conjectures (much of which is now theorem) precisely concern when the higher Tor terms behave. "You always need to move the cycles into general position." The classical moving lemma works for smooth quasi-projective varieties, but it can fail or be awkward (singular ambient, non-quasi-projective, families). Deformation to the normal cone constructs

V \cdot W

intrinsically without moving anything — this is the whole point of the Fulton–MacPherson reformulation, and it is what lets intersection theory function on singular spaces and in families.

Bézout fails without the three corrections

Take two lines in the affine plane $A_{R}^{2}$ : $y = x$ and $y = x + 1$ . They are degree-1 curves, so Bézout predicts $1 \cdot 1 = 1$ intersection point. But they are parallel and meet nowhere in $A^{2}$ . The failure has nothing to do with multiplicity — it is that affine space loses points at infinity. Passing to $P^{2}$ , the lines meet at the point $[1 : 1 : 0]$ on the line at infinity, restoring the count to $1$ . Lesson: Bézout [5] requires the projective, algebraically closed, multiplicity-weighted setting; drop any one and the count breaks.

Excess intersection: when $V\cap W$ is too big

In $P^{3}$ , let $V = W$ be the same line $L$ . Naively " $L \cdot L$ " should be a $dim L + dim L - 3 = 1 + 1 - 3 = - 1$ dimensional cycle — impossible. The intersection is not proper: $L \cap L = L$ has dimension $1$ , not $- 1$ . Naive multiplicity is undefined. The Fulton–MacPherson machinery still returns a well-defined class via the normal cone, but it lives in $A_{*} (L)$ and involves the excess normal bundle; the "expected" number must be corrected by a Segre/Chern class term. This is the prototype for the excess intersection formula.

Connections

Intersection theory sits at a crossroads. The Chow ring $A^{*} (X)$ is the algebraic-geometric sibling of singular cohomology $H^{*} (X)$ ; the cycle class map $A^{k} (X) \to H^{2 k} (X)$ links them and is the gateway to the Hodge Conjecture. The Gysin/pullback formalism is exactly that of Characteristic Classes — indeed Chern classes of vector bundles are defined via intersection theory, and the Grothendieck–Riemann–Roch Theorem expresses pushforward in $K$ -theory through Chern characters and Todd classes. The whole apparatus of rational equivalence parallels Singular Homology with principal divisors as boundaries, and it specializes for divisors to the theory of Divisors and Line Bundles and the Picard Group. In enumerative geometry it feeds Schubert Calculus on Grassmannians and modern Gromov–Witten Invariants.

Real-World Applications

Enumerative geometry / Schubert calculus. Counting lines on a cubic surface (the famous 27 lines), conics tangent to five given conics ( $3264$ , the problem of Apollonius generalized), and lines meeting four given lines in $P^{3}$ ( $2$ ) are all computed as intersection numbers in Chow rings of Grassmannians and Fano varieties.

Coding theory & finite geometry. Bézout-type bounds on the number of common zeros of polynomials over finite fields control the parameters of algebraic-geometric (Goppa) codes and underlie estimates for the number of $F_{q}$ -points on curves.

String theory & mirror symmetry. Intersection numbers on moduli spaces of stable maps are the Gromov–Witten invariants; computing them on Calabi–Yau threefolds (e.g. the quintic) yields the instanton numbers predicted by mirror symmetry.

Cryptography. Counting points on elliptic and hyperelliptic curves over finite fields (Schoof–Elkies–Atkin algorithm) rests on intersection-theoretic fixed-point formulas (Lefschetz/Weil), securing elliptic-curve cryptography.

Singularity theory & robotics. Milnor numbers and intersection multiplicities measure the local complexity of singularities, with applications to the topology of configuration spaces and kinematic singularities of robotic linkages.

Numerical algebraic geometry. Bézout and BKK (Bernstein–Khovanskii–Kushnirenko) bounds, the latter via Newton polytope mixed volumes, predict the number of solutions to polynomial systems, guiding homotopy continuation solvers in engineering design and chemical equilibrium computation.

Moduli & Witten's conjecture. Intersection numbers of psi-classes on $\overline{M}_{g, n}$ (Mumford–Morita–Miller classes) satisfy KdV-hierarchy relations (Kontsevich's theorem), connecting 2D quantum gravity to integrable systems.

Historical Development

1822 — Poncelet introduces the principle of continuity and points at infinity in projective geometry, the conceptual seed of "intersections that persist under deformation."
1779 / 1830s — Bézout's theorem, conjectured by Newton and proven by Étienne Bézout, asserts $d_{1} \dots d_{r}$ intersections — but multiplicity remains heuristic.
1864 — Chasles solves the conics problem ( $3264$ conics tangent to five), launching the enumerative geometry program and its many paradoxes.
1879 — Hermann Schubert publishes Kalkül der abzählenden Geometrie, a powerful but non-rigorous symbolic calculus; his methods motivate Hilbert's 15th problem: rigorize Schubert calculus.
1920s–30s — van der Waerden and the Italian school (Severi) push for rigor; van der Waerden gives early correct multiplicity definitions; Severi's intuition outruns available foundations.
1946 — André Weil's Foundations of Algebraic Geometry provides the first systematic rigorous intersection theory over arbitrary fields, using specialization and generic points.
1956 — Chevalley & Samuel give a local-algebra definition of multiplicity; Serre (1957–58) introduces the Tor-formula [3] and frames the multiplicity conjectures (vanishing, nonnegativity, positivity).
1956 — Chow defines what is now the Chow ring; rational equivalence becomes the canonical equivalence.
1981–1984 — Fulton & MacPherson develop deformation to the normal cone and the bivariant framework; Fulton's Intersection Theory (1984) becomes definitive, valid for singular schemes and in families.
1991 onward — Kontsevich, Witten, Vakil apply intersection theory on moduli spaces (psi-classes, virtual fundamental classes) to physics and Gromov–Witten theory.

Subtleties & Limits

Several deep questions remain partly open. Serre's positivity conjecture — that $i (Z; V, W) > 0$ whenever $V$ and $W$ meet properly along $Z$ in a regular local ring — was open for decades; the equicharacteristic and mixed-characteristic cases were resolved (Gabber, Roberts) only with heavy machinery, and the cleanest general statement still attracts work. The Chow groups of $0$ -cycles on a surface (Mumford's theorem: $A_{0} (X)$ can be infinite-dimensional) reveal that rational equivalence is vastly subtler than homological equivalence; the Bloch–Beilinson conjectures predict a filtration on Chow groups governed by Hodge theory and remain wide open. Whether the cycle class map is injective/surjective in suitable settings is the content of the Hodge Conjecture and the standard conjectures of Grothendieck. Finally, algebraic equivalence vs. rational equivalence vs. numerical equivalence differ in general, and understanding the precise relationships (e.g. the Griffiths group of cycles algebraically but not rationally trivial) is an active frontier.