04.05.06 · algebraic-geometry / surfaces

Intersection pairing on a surface

shipped3 tiersLean: nonepending prereqs

Anchor (Master): Hartshorne §V.1; Beauville *Complex Algebraic Surfaces* §I–§III; Fulton *Intersection Theory* Ch. 7–8; Zariski 1958 *Introduction to the Problem of Minimal Models*

Intuition [Beginner]

The intersection pairing on a surface is the rule that says, given two curves drawn on a smooth two-dimensional algebraic variety, you can count the number of points where they cross — and that count is a well-defined integer that does not change when you wiggle either curve. Two distinct lines in the projective plane meet at exactly one point. A line and a conic meet at two points (counted with multiplicity, a tangency contributing two). Two general conics in the plane meet at four points. The intersection number is the algebraic packaging of this counting.

Why should this be useful? On a smooth surface every curve has a divisor class — an equivalence class under linear equivalence — and the intersection number depends only on the class, not on the specific curves chosen as representatives. So the count of intersections becomes a bilinear pairing on the group of divisor classes. The pairing is symmetric: the number of crossings of $C$ with $C^{'}$ equals the number of crossings of $C^{'}$ with $C$ . And it is the foundational invariant that controls every classical geometric question on the surface, from genus formulas to embedding criteria to the classification of surfaces by Kodaira dimension.

The depth comes from the self-intersection number $D \cdot D$ , which counts how a curve crosses a slightly perturbed copy of itself. A line in the plane has self-intersection $1$ . A conic has self-intersection $4$ . The exceptional divisor of a blow-up at a point has self-intersection $- 1$ — the famous negative number that detects "extra" geometry sitting over the blown-up point. These three numbers — $1$ , $4$ , $- 1$ — are the first calculations every algebraic geometer learns about the intersection pairing.

Visual [Beginner]

A schematic of a smooth projective surface with two curves $C$ and $C^{'}$ drawn on it, meeting transversally at three points marked with crosses. Each crossing contributes $+ 1$ to the intersection number $C \cdot C^{'}$ . A second panel shows the same surface with the exceptional divisor $E$ of a blow-up at a point, depicted as a small projective line meeting itself in a "phantom" copy that contributes $E^{2} = - 1$ .

The picture captures the essential geometry: the intersection number is a count of crossings of two curves on a surface, with the convention that tangencies count multiply and self-intersections count via a perturbed copy of the curve. The same picture in higher dimensions replaces the curves with subvarieties of complementary dimension on a higher-dimensional variety.

Worked example [Beginner]

Compute three intersection numbers on three different surfaces, with specific divisors, and check that the answers match the geometry.

Step 1. The projective plane $P^{2}$ over an algebraically closed field. Every divisor on $P^{2}$ is linearly equivalent to a multiple of the hyperplane class $L$ , the class of any line. So divisors on $P^{2}$ form a copy of $Z$ generated by $L$ . The intersection $L \cdot L$ counts the number of points where two distinct lines meet. Two distinct lines in $P^{2}$ meet at exactly one point (in projective coordinates the two equations $a_{0} x_{0} + a_{1} x_{1} + a_{2} x_{2} = 0$ and $b_{0} x_{0} + b_{1} x_{1} + b_{2} x_{2} = 0$ have a one-dimensional null space, that is, a single projective point). So $L \cdot L = 1$ .

Step 2. Compute $C \cdot L$ for $C$ a smooth conic on $P^{2}$ , for instance the conic $x_{0}^{2} + x_{1}^{2} - x_{2}^{2} = 0$ . The conic $C$ is linearly equivalent to $2 L$ (every conic is a quadric, the divisor of a degree- $2$ polynomial). Bilinearity gives $C \cdot L = 2 L \cdot L = 2 \cdot 1 = 2$ . Geometrically: a generic line meets a smooth conic at two points, the standard fact that a quadratic equation has two roots.

Step 3. Compute $C \cdot C$ for the same conic. By bilinearity, $C \cdot C = 2 L \cdot 2 L = 4 L \cdot L = 4 \cdot 1 = 4$ . Geometrically: two general conics meet at four points (Bézout's theorem in $P^{2}$ for two degree- $2$ curves). The self-intersection $C^{2} = 4$ records this.

Step 4. The product surface $P^{1} \times P^{1}$ . The Picard group is $Z^{2}$ generated by the horizontal ruling $h$ (the class of ${point} \times P^{1}$ ) and the vertical ruling $f$ (the class of $P^{1} \times {point}$ ). The intersection numbers are $h \cdot h = 0$ (two horizontal rulings are parallel and disjoint), $f \cdot f = 0$ (two vertical rulings are parallel and disjoint), and $h \cdot f = 1$ (a horizontal and a vertical ruling meet at one point). The intersection matrix is $(0110)$ .

Step 5. The blow-up of $P^{2}$ at a point. The exceptional divisor $E$ replacing the blown-up point has self-intersection $E^{2} = - 1$ . Geometrically, $E$ is a copy of $P^{1}$ , and it meets a "perturbed" copy of itself in $- 1$ points — the negative number coming from the fact that the perturbed copy must come down off the exceptional divisor, contributing a negative correction. This $- 1$ is the signature of a blow-up.

What this tells us: the intersection numbers of the three surfaces above record three distinct geometric phenomena. On $P^{2}$ the pairing on the Picard group is rank one, with the single basis element $L$ having $L^{2} = 1$ . On $P^{1} \times P^{1}$ the pairing has rank two, with the two ruling classes $h$ , $f$ each having self-intersection zero and pairing to $1$ between them. On the blow-up $Bl_{p} P^{2}$ the new generator $E$ contributes a negative self-intersection $E^{2} = - 1$ , signalling that $E$ is contractible — and Castelnuovo's criterion later promotes this observation into a structural theorem.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

The intersection pairing on a smooth projective surface $X$ is a map:

A. $X \times X \to Z$ counting points
B. $Pic (X) \times Pic (X) \to Z$ that is symmetric and bilinear
C. $Pic (X) \to Z$ assigning a degree
D. $X \to X$ that swaps two curves

Hint

The pairing takes two divisor classes (or line bundles) and produces an integer. It is well-defined on linear-equivalence classes and is symmetric in its two arguments.

Answer

B. The intersection pairing is the symmetric bilinear map $Pic (X) \times Pic (X) \to Z$ , $(D, D^{'}) \mapsto D \cdot D^{'}$ , defined geometrically by counting transverse intersection points after replacing $D$ and $D^{'}$ by linearly-equivalent divisors meeting transversally, or equivalently via Euler characteristics of structure sheaves. The pairing depends only on the linear-equivalence class of each divisor, hence descends to the Picard group.

Formal definition [Intermediate+]

Let $k$ be an algebraically closed field and let $X$ be a smooth projective surface over $k$ , meaning $X$ is a smooth projective variety of dimension $2$ . Write $Div (X)$ for the group of Weil divisors and $Pic (X)$ for the Picard group of line bundles up to isomorphism. The two are identified on a smooth surface by the standard correspondence $D \mapsto O_{X} (D)$ .

Definition (intersection pairing). The intersection pairing on $X$ is the unique symmetric bilinear map $$ (,\cdot,) : \mathrm{Pic}(X) \times \mathrm{Pic}(X) \to \mathbb{Z}, \qquad (D, D') \mapsto D \cdot D' $$ satisfying the following two properties:

Geometric definition. If $C$ and $C^{'}$ are smooth curves on $X$ meeting transversally at finitely many points, then $C \cdot C^{'} = # (C \cap C^{'})$ , the number of intersection points.
Cohomological definition. For any two divisors $D, D^{'}$ on $X$ , $D \cdot D^{'} = χ (O_{X}) - χ (O_{X} (- D)) - χ (O_{X} (- D^{'})) + χ (O_{X} (- D - D^{'})),$ where $χ (F) = \sum_{i = 0}^{2} (- 1)^{i} dim_{k} H^{i} (X, F)$ is the Euler characteristic.

The cohomological formula is well-defined for all divisors (no transversality hypothesis needed); the geometric formula is recovered from it on transverse curves; the moving lemma on a smooth surface guarantees that every divisor class has a representative meeting a given divisor properly, so the cohomological and geometric definitions agree.

Definition (self-intersection). For a divisor $D$ on $X$ , the self-intersection number is $D^{2} = D \cdot D$ . Equivalently, $D^{2}$ is the cohomological invariant $$ D^2 = \chi(\mathcal{O}_X) - 2\chi(\mathcal{O}_X(-D)) + \chi(\mathcal{O}_X(-2D)). $$ Geometrically, $D^{2}$ is the intersection of $D$ with a divisor linearly equivalent to $D$ but distinct from $D$ — the "moved copy" — which is what makes self-intersection well-defined.

Definition (Néron-Severi group). The Néron-Severi group $NS (X)$ is the quotient of $Pic (X)$ by the connected component $Pic^{0} (X)$ . The intersection pairing factors through $NS (X)$ , since $Pic^{0}$ acts as zero on the pairing. The rank of $NS (X)$ is the Picard number $ρ (X)$ , a finite positive integer.

Definition (cycle-class map). The cycle-class map $cl : Pic (X) \to H^{2} (X, Z)$ sends a divisor class to its first Chern class, equivalently to its Poincaré dual under the identification $H^{2} (X, Z) ≅ H_{2} (X, Z)$ on a smooth complex projective surface. The intersection pairing on $Pic (X)$ corresponds under $cl$ to the cup product $⌣: H^{2} (X, Z) \otimes H^{2} (X, Z) \to H^{4} (X, Z) ≅ Z$ , evaluated against the fundamental class $[X]$ . Over a general algebraically closed field, the same identification holds with $ℓ$ -adic étale cohomology in place of singular cohomology.

Counterexamples to common slips

The intersection pairing is bilinear over $Z$ , not over $Z / n$ . Reducing the pairing mod $n$ for varying $n$ produces a family of bilinear forms on $NS (X) \otimes Z / n$ , but the pairing itself is integer-valued and integral coefficients matter.
The geometric definition requires transversality. For non-transverse curves, the naive count of intersection points is wrong — a tangency must contribute multiplicity, and the cohomological definition is the canonical extension. The moving lemma is what guarantees a transverse representative exists in the linear-equivalence class.
The self-intersection $D^{2}$ is not the geometric "number of times $D$ crosses itself" — a curve crosses itself zero times in the naive sense. Instead $D^{2}$ is the intersection of $D$ with a linearly equivalent moved copy, which can be negative for a curve like the exceptional divisor of a blow-up.
The pairing is symmetric and bilinear but not in general definite. On $P^{2}$ the pairing is positive definite (one positive eigenvalue), on $P^{1} \times P^{1}$ it has signature $(1, 1)$ , and on the blow-up $Bl_{p} P^{2}$ it has signature $(1, 1)$ as well. The Hodge index theorem (a downstream unit) gives the general signature.

Key theorem with proof [Intermediate+]

Theorem (well-definedness of the intersection pairing; Hartshorne V Theorem 1.1). Let $X$ be a smooth projective surface over an algebraically closed field $k$ . The map $$ \mathrm{Div}(X) \times \mathrm{Div}(X) \to \mathbb{Z}, \qquad (D, D') \mapsto \chi(\mathcal{O}_X) - \chi(\mathcal{O}_X(-D)) - \chi(\mathcal{O}_X(-D')) + \chi(\mathcal{O}_X(-D - D')) $$ is bilinear, symmetric, and depends only on the linear-equivalence classes of $D$ and $D^{'}$ . Moreover, when $C$ and $C^{'}$ are smooth curves on $X$ meeting transversally, the value of the map equals the number $# (C \cap C^{'})$ of intersection points.

Proof. The argument has three steps. First, establish that the cohomological formula descends to linear-equivalence classes. Second, prove bilinearity and symmetry. Third, verify the geometric interpretation on transverse curves.

Step 1: descent to linear equivalence. Linearly equivalent divisors $D \sim D^{'}$ correspond to isomorphic line bundles $O_{X} (D) ≅ O_{X} (D^{'})$ , and isomorphic line bundles have equal Euler characteristics: $χ (O_{X} (- D)) = χ (O_{X} (- D^{'}))$ . Replacing $D$ by a linearly equivalent divisor $D^{''}$ in the cohomological formula leaves all four Euler characteristics unchanged, so the value $D \cdot D^{'}$ depends only on the class $[D] \in Pic (X)$ . The same argument applies in the second slot.

Step 2: bilinearity and symmetry. Symmetry is immediate from the formula: swapping $D$ and $D^{'}$ leaves the four-term combination invariant. For bilinearity, fix $D^{'}$ and consider $D \mapsto D \cdot D^{'}$ as a function $Pic (X) \to Z$ . The key calculation: for divisors $D_{1}, D_{2}$ , we want to show $(D_{1} + D_{2}) \cdot D^{'} = D_{1} \cdot D^{'} + D_{2} \cdot D^{'}$ .

Expanding the cohomological formula on the left, $$ (D_1 + D_2) \cdot D' = \chi(\mathcal{O}_X) - \chi(\mathcal{O}_X(-D_1 - D_2)) - \chi(\mathcal{O}_X(-D')) + \chi(\mathcal{O}_X(-D_1 - D_2 - D')). $$ The right side is $$ [\chi(\mathcal{O}_X) - \chi(\mathcal{O}_X(-D_1)) - \chi(\mathcal{O}_X(-D')) + \chi(\mathcal{O}_X(-D_1 - D'))] + [\chi(\mathcal{O}_X) - \chi(\mathcal{O}_X(-D_2)) - \chi(\mathcal{O}_X(-D')) + \chi(\mathcal{O}_X(-D_2 - D'))]. $$

The difference between the two sides equals $$ 2\chi(\mathcal{O}_X) - \chi(\mathcal{O}_X(-D_1)) - \chi(\mathcal{O}_X(-D_2)) - \chi(\mathcal{O}_X(-D')) - \chi(\mathcal{O}_X(-D_1 - D_2)) + \chi(\mathcal{O}_X(-D_1 - D')) + \chi(\mathcal{O}_X(-D_2 - D')) - \chi(\mathcal{O}_X(-D_1 - D_2 - D')). $$

This eight-term combination vanishes by an additivity calculation on the Euler characteristic. The shortest route uses the short exact sequence $$ 0 \to \mathcal{O}_X(-D_1 - D_2) \to \mathcal{O}_X(-D_1) \oplus \mathcal{O}_X(-D_2) \to \mathcal{O}_X \to 0 $$ of line bundles, which (after twisting by $O_{X} (- D^{'})$ where appropriate) gives the additivity relation needed. The detailed Euler-characteristic bookkeeping is in Hartshorne V Lemma 1.3 ^[pending].

Step 3: geometric interpretation. Suppose $C$ and $C^{'}$ are smooth curves on $X$ meeting transversally at points $p_{1}, \dots, p_{n}$ . The exact sequence $$ 0 \to \mathcal{O}_X(-C - C') \to \mathcal{O}_X(-C) \oplus \mathcal{O}_X(-C') \to \mathcal{O}X \to \mathcal{O}{C \cap C'} \to 0 $$ of $O_{X}$ -modules, where $C \cap C^{'}$ is the scheme-theoretic intersection (a length- $n$ zero-dimensional subscheme since $C$ and $C^{'}$ are transverse), is exact. Taking Euler characteristics and using additivity along the sequence, $$ \chi(\mathcal{O}_X(-C - C')) - \chi(\mathcal{O}_X(-C)) - \chi(\mathcal{O}X(-C')) + \chi(\mathcal{O}X) = \chi(\mathcal{O}{C \cap C'}) = n, $$ where the last equality holds because $\mathcal{O}{C \cap C'} $i s a l e n g t h -$ n $s k y scr a p er s h e a f an d i t s E u l er c ha r a c t er i s t i ce q u a l s t h e n u mb er o f p o in t s . R e a r r an g in g g i v es$ C \cdot C' = n $.$ \square$

Bridge. The intersection pairing builds toward the surface-classification theory that Castelnuovo, Enriques, and Beauville assembled across the late nineteenth and twentieth centuries. The central insight is that on a smooth projective surface, the bilinear pairing $Pic (X) \times Pic (X) \to Z$ — defined geometrically as a count of transverse intersections, or cohomologically as a four-term combination of Euler characteristics — captures every classical numerical invariant, and the two definitions agree by the moving lemma. This is exactly the bridge that appears again in 04.05.07 (adjunction formula), where the genus of a smooth curve $C$ on $X$ is computed by the formula $2 g - 2 = K_{X} \cdot C + C^{2}$ , and in 04.05.08 (Riemann-Roch for surfaces), where the Euler characteristic of a line bundle is $χ (O_{X} (D)) = χ (O_{X}) + \frac{1}{2} D \cdot (D - K_{X})$ . Both successor results depend on the intersection pairing as the foundational bilinear invariant, and both are corollaries of the cohomological formula together with Serre duality. Putting these together, the Hartshorne §V.1 framework gives the foundational invariants of surface theory in three theorems: well-definedness of the intersection pairing, the adjunction formula, and Riemann-Roch for surfaces, with the Hodge index theorem in 04.05.09 pending supplying the signature data on $NS (X) \otimes R$ .

Exercises [Intermediate+]

Exercise 5 (medium, short-answer).

Show that on the blow-up $X = Bl_{p} X$ of a smooth surface $X$ at a point $p$ , with exceptional divisor $E$ and total transform $π^{*} D$ of a divisor $D$ on $X$ , the intersection numbers satisfy $E \cdot π^{*} D = 0$ and $E^{2} = - 1$ .

Hint

Move $D$ off the point $p$ by linear equivalence, so that $π^{*} D$ is supported away from $E$ . For $E^{2}$ , use the projection formula or the local computation via the conormal sheaf $N_{E / X}^{\lor} = O_{E} (1)$ .

Answer

For $E \cdot π^{*} D$ : replace $D$ by a linearly equivalent divisor $D^{'}$ not passing through $p$ . Then $π^{*} D^{'}$ is the strict transform $D^{'}$ of $D^{'}$ , which lies entirely above $X ∖ {p}$ and is therefore disjoint from $E$ . Disjoint divisors have intersection $0$ , so $E \cdot π^{*} D = E \cdot π^{*} D^{'} = 0$ .

For $E^{2}$ : the exceptional divisor $E$ of the blow-up at a smooth point of a surface is isomorphic to $P^{1}$ , and its conormal sheaf in $X$ is $N_{E / X}^{\lor} = O_{E} (1)$ — the dual of the tautological line bundle on $P^{1}$ , which has degree $1$ . By the standard self-intersection formula for a curve $E$ on a surface $X$ , $E^{2} = de g (N_{E / X}) = - de g (N_{E / X}^{\lor}) = - 1$ . Rubric: full credit for both intersection numbers with the disjointness and conormal-sheaf arguments.

Exercise 7 (hard, short-answer).

State and prove the projection formula for the intersection pairing under a proper morphism $f : X \to Y$ of smooth projective surfaces: $f_{*} (f^{*} D \cdot D^{'}) = D \cdot f_{*} D^{'}$ for divisors $D$ on $Y$ and $D^{'}$ on $X$ , where $f_{*}$ on divisors is push-forward and $f^{*}$ is pullback.

Hint

Use the geometric definition: replace $D$ and $D^{'}$ by linearly equivalent transverse divisors, and check the formula geometrically. Alternatively use the cohomological definition and the projection formula for sheaves $f_{*} (f^{*} F \otimes G) = F \otimes f_{*} G$ .

Answer

By the moving lemma, replace $D$ on $Y$ by a linearly equivalent transverse divisor avoiding the branch locus of $f$ , and replace $D^{'}$ on $X$ by a transverse divisor for which $f ∣_{D^{'}} : D^{'} \to f (D^{'})$ is generically étale. Then $f^{*} D$ and $D^{'}$ meet transversally on $X$ , and the points of $f^{*} D \cap D^{'}$ map under $f$ to the points of $D \cap f_{*} D^{'}$ on $Y$ , with the multiplicity at each point equal to the local degree of $f$ .

For the cohomological proof: by the projection formula for sheaves, $$ f_*(\mathcal{O}_X(f^(-D)) \otimes \mathcal{O}_X(-D')) = \mathcal{O}Y(-D) \otimes f \mathcal{O}_X(-D'). $$ Taking Euler characteristics and using that $χ$ is invariant under proper push-forward (the Leray spectral sequence collapses for $f$ proper between smooth projective varieties of the same dimension up to a degree factor), $$ \chi(\mathcal{O}X(f^(-D) - D')) = \chi(\mathcal{O}Y(-D) \otimes f \mathcal{O}X(-D')). $$ Substituting into the four-term cohomological formula for $f^{*} D \cdot D^{'}$ and unwinding gives $f(f^ D \cdot D') = D \cdot f* D'$. The general projection formula in intersection theory is Fulton Theorem 1.4 ^[pending]. Rubric: full credit for the geometric and cohomological formulations with the local-degree counting.

Exercise 8 (hard, short-answer).

The cubic surface $S \subset P^{3}$ contains exactly $27$ lines. Show that the $27$ lines on $S$ all have self-intersection $- 1$ , and describe the intersection matrix.

Hint

A cubic surface is isomorphic to $P^{2}$ blown up at $6$ general points. The $27$ lines correspond to: the $6$ exceptional divisors, the $15$ proper transforms of lines through pairs of the blown-up points, and the $6$ proper transforms of conics through five of the six points.

Answer

Let $X = Bl_{p_{1}, \dots, p_{6}} P^{2}$ , with exceptional divisors $E_{1}, \dots, E_{6}$ and $L$ the pullback of the hyperplane class. The Picard group is $Z^{7}$ with intersection numbers $L^{2} = 1$ , $E_{i}^{2} = - 1$ , $L \cdot E_{i} = 0$ , $E_{i} \cdot E_{j} = 0$ for $i \neq = j$ .

The $27$ lines on the cubic are, with classes:

$E_{i}$ for $i = 1, \dots, 6$ (the $6$ exceptional divisors): $E_{i}^{2} = - 1$ .
$L_{ij} = L - E_{i} - E_{j}$ for $1 \leq i < j \leq 6$ (the $15$ proper transforms of lines through $p_{i}$ and $p_{j}$ ): $L_{ij}^{2} = L^{2} - 2 L \cdot (E_{i} + E_{j}) + (E_{i} + E_{j})^{2} = 1 - 0 + (E_{i}^{2} + 2 E_{i} \cdot E_{j} + E_{j}^{2}) = 1 - 2 = - 1$ .
$C_{i} = 2 L - E_{1} - \dots - E_{i} - \dots - E_{6}$ for $i = 1, \dots, 6$ (the $6$ proper transforms of conics through five of the six points): $C_{i}^{2} = 4 L^{2} - 2 \cdot 2 L \cdot \sum_{j \neq = i} E_{j} + (\sum_{j \neq = i} E_{j})^{2} = 4 - 0 + 5 \cdot (- 1) = 4 - 5 = - 1$ .

Total: $6 + 15 + 6 = 27$ lines, all of self-intersection $- 1$ . The intersection matrix encodes the $E_{6}$ root system: the orthogonal complement of the canonical class $K = - 3 L + E_{1} + \dots + E_{6}$ in the Picard lattice is the $E_{6}$ root lattice, and the $27$ lines are the weights of the $27$ -dimensional representation of $E_{6}$ . This is one of the most famous incidences of algebraic geometry. Rubric: full credit for the self-intersection computation in each of the three classes plus identification of the count.

Lean formalization [Intermediate+]

Mathlib has the divisor and Picard-group infrastructure for smooth varieties but no named intersection pairing on a surface. The intended formalisation reads schematically:

[object Promise]

The proof gap is substantive. Mathlib needs the Euler-characteristic formalism on smooth projective surfaces specialised to dimension $2$ , the additivity of $χ$ along the canonical four-term sequence on line bundles, the moving lemma for divisors on a surface, and the transverse-intersection comparison via length of the scheme-theoretic intersection. Each piece is formalisable from existing infrastructure but has not been packaged. The bilinearity proof (the eight-term cancellation in Step 2 of the main theorem) is the natural first formalisation target, and would deliver a Mathlib-grade intersection pairing on smooth projective surfaces.

Advanced results [Master]

Theorem (cohomological formula; Hartshorne V Lemma 1.4). On a smooth projective surface $X$ over an algebraically closed field $k$ , the intersection pairing $D \cdot D^{'}$ admits the cohomological description $$ D \cdot D' = \chi(\mathcal{O}_X) - \chi(\mathcal{O}_X(-D)) - \chi(\mathcal{O}_X(-D')) + \chi(\mathcal{O}_X(-D - D')). $$ The four-term combination is the second discrete derivative of $χ (O_{X} (- a D - b D^{'}))$ as a polynomial in $(a, b)$ , evaluated at $(0, 0)$ .

The polynomial structure: by Riemann-Roch on a surface, $χ (O_{X} (D))$ is a quadratic polynomial in the divisor class $D \in Pic (X) \otimes Q$ . The four-term combination is exactly the discrete bilinear-form extraction from a quadratic polynomial, which is what makes the cohomological definition produce a bilinear symmetric pairing.

Theorem (cup-product compatibility). Let $X$ be a smooth projective surface over $C$ . The cycle-class map $cl : Pic (X) \to H^{2} (X, Z)$ intertwines the intersection pairing on $Pic (X)$ with the cup product followed by evaluation against the fundamental class $[X] \in H_{4} (X, Z)$ : $$ D \cdot D' = \langle \mathrm{cl}(D) \smile \mathrm{cl}(D'), [X] \rangle. $$ Over a general algebraically closed field $k$ , the same identification holds with $ℓ$ -adic étale cohomology in place of singular cohomology, with the cycle-class map landing in $H_{\overset{e}{ˊ} t}^{2} (X, Z_{ℓ} (1))$ .

This is the topological face of the intersection pairing. On a smooth complex projective surface, the pairing is the restriction of the cup-product pairing on $H^{2} (X, Z)$ to the image of $Pic (X)$ , and the lattice-theoretic structure of the pairing — its rank, signature, and discriminant — is read off from singular cohomology.

Theorem (adjunction formula; Hartshorne V Proposition 1.5). Let $C$ be a smooth curve on a smooth projective surface $X$ , with canonical divisor $K_{X}$ and genus $g (C)$ . Then $$ 2 g(C) - 2 = (K_X + C) \cdot C = K_X \cdot C + C^2. $$ Equivalently, the canonical line bundle of $C$ is $ω_{C} ≅ (ω_{X} \otimes O_{X} (C)) ∣_{C}$ .

This is the foundational genus-computing formula for curves on surfaces. A line $L$ on $P^{2}$ has $L^{2} = 1$ and $K \cdot L = - 3$ , giving $2 g - 2 = - 3 + 1 = - 2$ , hence $g = 0$ — the line is rational, as expected. A smooth quartic on $P^{2}$ has $Q^{2} = 16$ , $K \cdot Q = - 12$ , giving $2 g - 2 = 4$ , hence $g = 3$ — the canonical curve of genus $3$ .

Theorem (Riemann-Roch for surfaces; Hartshorne V Theorem 1.6). For a divisor $D$ on a smooth projective surface $X$ over an algebraically closed field, $$ \chi(\mathcal{O}_X(D)) = \chi(\mathcal{O}_X) + \tfrac{1}{2} D \cdot (D - K_X). $$ Equivalently, $χ (O_{X} (D)) = \frac{1}{2} (D^{2} - D \cdot K_{X}) + χ (O_{X})$ .

The intersection pairing is the input to the Riemann-Roch formula on a surface. The two terms on the right — the quadratic form $\frac{1}{2} D^{2}$ and the linear functional $- \frac{1}{2} D \cdot K_{X}$ — are exactly the data extracted from a quadratic polynomial in $D$ , with the constant term $χ (O_{X})$ recording the underlying surface invariants.

Theorem (Hodge index theorem; Hartshorne V Theorem 1.9). Let $X$ be a smooth projective surface over an algebraically closed field. The intersection pairing on $NS (X) \otimes R$ has signature $(1, ρ - 1)$ , where $ρ = rk NS (X)$ is the Picard number.

The signature theorem is the deep lattice-theoretic structure underlying the pairing. The single positive direction is spanned by an ample class; the $ρ - 1$ negative directions are the orthogonal complement. On $P^{2}$ the Picard number is $1$ and the signature is $(1, 0)$ . On $P^{1} \times P^{1}$ and $Bl_{p} P^{2}$ the Picard number is $2$ and the signature is $(1, 1)$ . On the cubic surface ( $P^{2}$ blown up at $6$ general points) the Picard number is $7$ and the signature is $(1, 6)$ , with the negative-definite orthogonal complement of the canonical class being the $E_{6}$ root lattice.

Theorem (Castelnuovo's contractibility criterion; Hartshorne V Theorem 5.7). A smooth rational curve $E \subset X$ on a smooth projective surface $X$ with $E^{2} = - 1$ is contractible: there is a smooth projective surface $X^{'}$ and a morphism $π : X \to X^{'}$ realising $X$ as the blow-up of $X^{'}$ at a smooth point with exceptional divisor $E$ .

The negative self-intersection $E^{2} = - 1$ together with rationality is the precise diagnostic for an exceptional divisor of a blow-up. Castelnuovo's theorem is the converse to the blow-up calculation: every smooth rational curve with $E^{2} = - 1$ on a surface arises as the exceptional divisor of a contraction. This is the input to the minimal model program for surfaces.

Theorem (intersection-form classification of surfaces; Castelnuovo-Beauville-Bombieri-Kodaira). The Kodaira dimension $κ (X) \in {- \infty, 0, 1, 2}$ of a smooth projective surface determines the asymptotic behaviour of the canonical ring $⨁_{n} H^{0} (X, n K_{X})$ , which in turn is read off the intersection pairing $K_{X}^{2}$ together with the canonical bundle's positivity.

The full classification: $κ = - \infty$ (rational and ruled surfaces), $κ = 0$ (K3, abelian, Enriques, hyperelliptic surfaces), $κ = 1$ (properly elliptic surfaces), $κ = 2$ (surfaces of general type). The intersection theory on $X$ — the values of $K_{X}^{2}$ , the Picard number $ρ$ , the signature of the pairing — is what distinguishes the cases. Beauville's Complex Algebraic Surfaces §III–§X is the canonical worked-through treatment.

Synthesis. The intersection pairing is the foundational invariant of a smooth projective surface, and the central insight is that the bilinear pairing $Pic (X) \times Pic (X) \to Z$ — defined geometrically by counting transverse intersections, cohomologically by a four-term combination of Euler characteristics, and topologically by the cup product on second cohomology — captures every classical numerical invariant of the surface and unifies three apparently distinct constructions into one symmetric bilinear form. Putting these together, the intersection pairing is what makes the adjunction formula compute genera of embedded curves, what makes Riemann-Roch on a surface produce dimensions of line-bundle cohomology, what makes the Hodge index theorem identify the signature on the Néron-Severi group, and what makes Castelnuovo's contractibility criterion detect blow-ups via the diagnostic $E^{2} = - 1$ . The bridge between the topological and the cohomological framings is the cycle-class map together with the projection formula, and the bridge between the geometric and the cohomological framings is the moving lemma. This is exactly the bridge that appears again in 04.05.07 (adjunction formula), where the genus computation $2 g - 2 = K \cdot C + C^{2}$ depends on the intersection pairing, and in 04.05.08 (Riemann-Roch for surfaces), where the Euler characteristic $χ (O_{X} (D)) = χ (O_{X}) + \frac{1}{2} D \cdot (D - K_{X})$ depends on the same pairing.

The intersection pairing also generalises in two directions. To higher dimensions, the pairing on $Pic (X)$ becomes the intersection product on the Chow ring $A^{*} (X)$ , with $A^{k} (X) \otimes A^{n - k} (X) \to A^{n} (X) ≅ Z$ extracting an integer from cycle classes of complementary dimension on a smooth projective $n$ -dimensional variety. Fulton's Intersection Theory (1984) is the canonical reference for the higher-dimensional theory, and the surface case is the prototype that motivates the Chow-ring construction. To $ℓ$ -adic étale cohomology, the cup product on $H_{\overset{e}{ˊ} t}^{*} (X, Z_{ℓ})$ provides the topological pairing in positive characteristic, and the Weil conjectures relate the trace of Frobenius on $NS (X) \otimes Q_{ℓ}$ to point counts on $X$ over finite fields. The classical Italian-school intersection pairing thus extends, via Grothendieck's étale cohomology, to the foundational tool of arithmetic geometry. The recursion stabilises after one round-trip: the intersection pairing on a surface is a special case of the Chow-ring intersection product, which is itself an algebraic incarnation of the cup product, which on a smooth projective complex variety recovers the intersection pairing through Poincaré duality.

Full proof set [Master]

Theorem (well-definedness), proof. Given in the Intermediate-tier section: descent to linear equivalence is immediate from the cohomological formula; bilinearity follows from an eight-term cancellation that uses the additivity of $χ$ along the short exact sequence $0 \to O_{X} (- D_{1} - D_{2}) \to O_{X} (- D_{1}) \oplus O_{X} (- D_{2}) \to O_{X} \to 0$ ; symmetry is immediate from the formula; the geometric interpretation on transverse smooth curves comes from the four-term Koszul-style exact sequence whose right end is $O_{C \cap C^{'}}$ , a length- $n$ skyscraper sheaf when the intersection is transverse. $□$

Theorem (cohomological formula), proof. Riemann-Roch on a surface, applied to $O_{X} (- a D - b D^{'})$ , gives $$ \chi(\mathcal{O}_X(-aD - bD')) = \chi(\mathcal{O}_X) + \tfrac{1}{2} (-aD - bD') \cdot (-aD - bD' - K_X). $$ Expanding the quadratic form in $(a, b)$ , $$ \chi(\mathcal{O}_X(-aD - bD')) = \chi(\mathcal{O}_X) + \tfrac{1}{2} (a^2 D^2 + 2ab D \cdot D' + b^2 (D')^2 + a D \cdot K_X + b D' \cdot K_X). $$ The four-term combination $$ \chi(\mathcal{O}_X) - \chi(\mathcal{O}_X(-D)) - \chi(\mathcal{O}_X(-D')) + \chi(\mathcal{O}_X(-D - D')) $$ extracts the coefficient of the cross-term $ab$ , which is $D \cdot D^{'}$ . The linear and quadratic terms in $a$ alone or $b$ alone cancel in the four-term combination (since they appear with coefficients $- 1 - 1 + 1 + 1 = 0$ when summed with appropriate signs). The constant term cancels for the same reason. So the four-term combination equals $D \cdot D^{'}$ , confirming the cohomological formula. $□$

Theorem (adjunction formula), proof. The conormal sheaf of $C$ in $X$ is $N_{C / X}^{\lor} = O_{X} (- C) ∣_{C}$ , so the normal sheaf is $N_{C / X} = O_{X} (C) ∣_{C}$ . The conormal exact sequence $$ 0 \to \mathcal{N}_{C/X}^\vee \to \Omega^1_X|_C \to \Omega^1_C \to 0 $$ is exact (since $C$ is smooth and $X$ is smooth). Taking determinants and using $det (Ω_{X}^{1}) = ω_{X}$ , $det (Ω_{C}^{1}) = ω_{C}$ , $$ \omega_C \cong \omega_X|C \otimes \mathcal{N}{C/X} \cong (\omega_X \otimes \mathcal{O}_X(C))|_C. $$ Taking degrees on $C$ and using $de g (ω_{C}) = 2 g (C) - 2$ , $de g (O_{X} (D) ∣_{C}) = D \cdot C$ , $$ 2 g(C) - 2 = K_X \cdot C + C \cdot C = K_X \cdot C + C^2. \qquad \square $$

Theorem (Riemann-Roch for surfaces), stated without proof here — full proof in the §V.1.6 of Hartshorne ^[pending]. The proof uses Serre duality on the surface, Riemann-Roch on smooth curves embedded in the surface, and an induction on the divisor class.

Theorem (Hodge index), stated without proof here — full proof in Hartshorne §V Theorem 1.9 ^[pending]. The proof uses the Lefschetz $(1, 1)$ theorem identifying $NS (X) \otimes R$ with the Lefschetz hyperplane in $H^{1, 1} (X, R)$ , and the standard signature computation of the cup-product form on $H^{1, 1}$ for a Kähler surface, which has signature $(1, h^{1, 1} - 1)$ by the Hodge-Riemann bilinear relations.

Theorem (Castelnuovo's criterion), stated without proof here — full proof in Hartshorne §V Theorem 5.7 ^[pending]. The proof constructs the contraction $π : X \to X^{'}$ via the linear system $∣ n L ∣$ for $L = K_{X} + n E$ with $n$ large enough that $L$ becomes globally generated and contracts $E$ to a point.

Theorem (cup-product compatibility), proof sketch. For a smooth projective complex surface $X$ , the cycle-class map $cl : Pic (X) \to H^{2} (X, Z)$ sends a divisor $D$ to its first Chern class $c_{1} (O_{X} (D))$ , equivalently to the Poincaré dual of $[D] \in H_{2} (X, Z)$ . For two transverse curves $C, C^{'}$ , the cup product $cl (C) ⌣ cl (C^{'}) \in H^{4} (X, Z) ≅ Z$ evaluated against $[X]$ counts the transverse intersection points $# (C \cap C^{'})$ by the Poincaré-Lefschetz transverse-intersection formula. So $cl (C) ⌣ cl (C^{'}) = (C \cdot C^{'}) \cdot [X]$ , and evaluating against $[X]$ gives the agreement of the cup-product pairing with the intersection pairing. The general case (non-transverse divisors) follows from the moving lemma. $□$

Connections [Master]

Picard group 04.05.02. The intersection pairing is a bilinear form on the Picard group, and the Picard group is the natural abelian-group setting in which the pairing is well-defined. Without the Picard-group identification of line bundles up to isomorphism, the pairing would be defined only on the larger divisor group with a nontrivial kernel of linear equivalence; the descent to $Pic (X)$ is what makes the pairing a clean numerical invariant.
Cartier divisor 04.05.04. Every divisor on a smooth projective surface is both Weil and Cartier (the surface is locally factorial), and the line bundle $O_{X} (D)$ associated to a Cartier divisor is what enters the cohomological formula. The Cartier-Weil agreement on smooth surfaces is the technical input that makes the cohomological definition coincide with the geometric definition.
Line bundle 04.05.03. The cohomological formula for $D \cdot D^{'}$ is a function of the line bundles $O_{X} (- D)$ and $O_{X} (- D - D^{'})$ , so the pairing factors through the line-bundle picture. The intersection pairing on a surface is, equivalently, a bilinear form on the abelian group of line bundles up to isomorphism.
Blowup 04.07.02. The blow-up of a smooth surface at a point introduces an exceptional divisor with self-intersection $E^{2} = - 1$ . This is the foundational example of a negative-self-intersection curve and the reason the intersection pairing is not positive definite in general. Castelnuovo's contractibility criterion is the converse: every smooth rational curve with $E^{2} = - 1$ is the exceptional divisor of a blow-up.
Sheaf cohomology 04.03.01. The cohomological formula for the intersection pairing depends on the Euler characteristic $χ (F) = \sum_{i} (- 1)^{i} dim H^{i} (X, F)$ , which is the alternating sum of dimensions of sheaf cohomology groups. Without sheaf cohomology, the cohomological definition of the pairing has no setting; with it, the definition is a four-term combination of Euler characteristics that is automatically well-defined and bilinear.
Canonical sheaf 04.08.02. The adjunction formula $2 g (C) - 2 = K_{X} \cdot C + C^{2}$ on a surface couples the intersection pairing with the canonical class $K_{X}$ , the divisor of the canonical sheaf. The pairing of the canonical class with itself, $K_{X}^{2}$ , is one of the fundamental numerical invariants of a surface and enters the surface classification.
Riemann-Roch theorem 04.04.01. The Riemann-Roch theorem for curves is the one-dimensional analogue of Riemann-Roch on a surface, which itself uses the intersection pairing as input. The cohomological formula for the intersection pairing on a surface is, after taking the four-term combination, a calculation entirely controlled by Riemann-Roch on the surface, illustrating the extent to which both theorems are different facets of the same calculation.
Poincaré duality 03.12.16. The cup-product compatibility of the intersection pairing with the cup product on $H^{2} (X, Z)$ is an instance of Poincaré duality on a smooth complex projective surface: the cup-product pairing $H^{2} (X) \otimes H^{2} (X) \to H^{4} (X) ≅ Z$ is the topological face of the geometric intersection pairing, and the cycle-class map intertwines the two. The intersection pairing on $Pic (X)$ is the algebraic-geometric incarnation of the topological intersection form on $H_{2} (X, Z)$ via Poincaré duality.

Historical & philosophical context [Master]

The intersection pairing on a surface was first studied by the Italian school of algebraic geometry: Guido Castelnuovo, Federigo Enriques, and Francesco Severi, working in Bologna, Pisa, and Rome between 1880 and 1910. Castelnuovo's Sulle superficie di genere zero (Mem. Soc. It. delle Scienze 1896) ^[pending] introduced the early intersection-form computations on rational surfaces, including the rationality criterion that bears his name. Enriques' Le Superficie Algebriche (Bologna 1949) ^[pending] is the synthesis of the Italian-school surface theory, with the intersection pairing playing the role of the foundational numerical invariant and the birational classification of surfaces (the "Enriques classification") proceeding by case analysis on the values of $K^{2}$ , $χ (O_{X})$ , and the Picard number. Severi's geometric definition of intersection multiplicities, refined into a moving lemma for cycles on a smooth projective variety, gave the geometric face of the pairing.

The modern scheme-theoretic framing was assembled by Oscar Zariski and Pierre Samuel in the 1940s and 1950s, and by André Weil in his Foundations of Algebraic Geometry (1946). Zariski's Introduction to the Problem of Minimal Models (Publ. Math. Soc. Japan 1958) ^[pending] gave the modern minimal-model program for surfaces using the intersection pairing as the foundational tool, and Robin Hartshorne's Algebraic Geometry (Springer GTM 52, 1977) §V is the canonical English-language treatment. Hartshorne presents the cohomological definition first, deduces the geometric interpretation as a corollary, and develops the adjunction formula and Riemann-Roch on a surface as immediate consequences. William Fulton's Intersection Theory (Springer Ergebnisse 2, 1984) ^[pending] is the higher-dimensional generalisation: the Chow ring $A^{*} (X)$ of a smooth projective variety, with intersection product $A^{k} (X) \otimes A^{n - k} (X) \to A^{n} (X) ≅ Z$ , contains the surface intersection pairing as the special case $n = 2$ , $k = 1$ .

The classification of smooth projective surfaces by Kodaira dimension — what is now called the Castelnuovo-Beauville-Bombieri-Kodaira classification — was completed in the 1960s and 1970s by Kunihiko Kodaira (in characteristic zero) and Enrico Bombieri together with David Mumford (in positive characteristic). Arnaud Beauville's Complex Algebraic Surfaces (Cambridge 1978) ^[pending] is the modern textbook treatment, with the intersection pairing as the foundational invariant and the four-class classification (rational, ruled; K3, abelian, Enriques, hyperelliptic; properly elliptic; general type) as the structural endpoint. The pairing extends to $ℓ$ -adic étale cohomology in positive characteristic, and Pierre Deligne's proof of the Weil conjectures uses the intersection pairing on Néron-Severi groups of surfaces over finite fields as a key input.

Bibliography [Master]

[object Promise]

Prerequisites

04.02.01 pending
04.05.01 pending
04.05.03 pending
04.05.04 pending
04.05.02 pending

Used in

04.05.07
04.05.08
04.05.09

Tier anchors

beginner: Beauville *Complex Algebraic Surfaces* §I — counting transverse intersection points on a surface
intermediate: Hartshorne *Algebraic Geometry* §V.1 (Theorem 1.1 well-definedness, V.1.4 cohomological formula); Beauville §I
master: Hartshorne §V.1; Beauville *Complex Algebraic Surfaces* §I–§III; Fulton *Intersection Theory* Ch. 7–8; Zariski 1958 *Introduction to the Problem of Minimal Models*

References

TODO_REF
Hartshorne — Algebraic Geometry · §V.1 (Theorem 1.1 — bilinear, symmetric, well-defined; Lemma 1.4 — cohomological formula via Euler characteristics)
TODO_REF
Beauville — Complex Algebraic Surfaces · §I (intersection form on a surface), §II (birational maps, blow-up self-intersection $E^2 = -1$), §III (ruled surfaces and Hirzebruch surfaces $\mathbb{F}_n$)
TODO_REF
Fulton — Intersection Theory · Ch. 7 (intersection on surfaces from the $A^*(X)$ Chow ring), Ch. 8 (refined Gysin homomorphisms, projection formula)
TODO_REF
Zariski — Introduction to the Problem of Minimal Models in the Theory of Algebraic Surfaces · Publ. Math. Soc. Japan 4 (1958); the modern intersection-theoretic framing of the Italian-school invariants
TODO_REF
Castelnuovo — Sulle superficie di genere zero · Mem. Soc. It. delle Scienze 10 (1896); originator-text for the rationality criterion and the early intersection-form computations
TODO_REF
Enriques — Le Superficie Algebriche · Bologna 1949; the Italian-school synthesis of surface theory, intersection form, and birational classification

Reviewer

TBD

Estimated time

beginner: 18m
intermediate: 45m
master: 75m