Intersection theory is one of the fastest ways to feel what algebraic geometry is doing behind the scenes. You start with a concrete question that sounds like classical geometry—how many \times do two curves meet?—and you end up with a toolkit that works in families, survives degenerations, and produces invariants that classify varieties.
The best way to learn it is through worked examples that repeat the same pattern:
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- translate geometry into divisors or cycles,
- replace “count” with “intersection number,” which remembers multiplicity,
- compute using line bundles, classes, and functoriality,
- interpret the answer geometrically.
This article runs that pattern several \times, each time with slightly richer structure, so you can see the thread clearly.
Example 1: two plane curves and why multiplicity is not optional
Let $C$ and $D$ be plane curves in $\mathbf{P}^2$ defined by homogeneous polynomials of degrees $m$ and $n$. Classically, you expect $mn$ intersection points. But that is not literally true as a set: curves can be tangent, share components, or meet at fewer points with higher order contact.
Intersection theory fixes the statement by upgrading “number of points” \to “number of points counted with multiplicity.”
At a point $p\in C\cap D$, define the local intersection multiplicity $I_p(C,D)$. One algebraic definition is:
when $f$ and $g$ are local equations of $C$ and $D$ in the local ring at $p$, and the intersection is proper near $p$.
This already teaches a key lesson: intersection multiplicity is not a geometric afterthought; it is an invariant of a local algebra.
When $C$ and $D$ meet transversely at a smooth point, $I_p(C,D)=1$. When they are tangent, the quotient ring grows and the multiplicity increases.
Bezout’s theorem as the first global computation
Bezout’s theorem states that if $C$ and $D$ have no common component, then
Notice the structure: a global invariant $mn$ equals a sum of local invariants. This “local-\to-global through a conservation law” is the same structural shape you see later in cohomology and Riemann–Roch.
A proof strategy perspective:
- local multiplicity is defined in commutative algebra,
- the global identity is proved using projective geometry and the behavior of divisors,
- the conclusion is stable under deformation: if you move one curve slightly, intersection points move but the total weighted count stays fixed.
Example 2: divisors and line bundles on $\mathbf{P}^2$
A divisor on a smooth variety is a formal integer combination of codimension-one subvarieties. On $\mathbf{P}^2$, every effective divisor of degree $d$ is linearly equivalent \to $dH$, where $H$ is the class of a line.
The Picard group is:
Intersection pairing on a smooth surface gives a bilinear map
and on $\mathbf{P}^2$ it is determined by
So if $C\sim mH$ and $D\sim nH$, then
This is Bezout’s theorem in a single line, once the language is set up. What looked like a geometric counting statement becomes an identity in the intersection ring.
The thread you should notice:
- you reduce geometry to classes in $\mathrm{Pic}$,
- you compute using bilinearity and a normalization $H\cdot H=1$,
- you interpret the answer back as a total multiplicity.
Example 3: $\mathbf{P}^1\times \mathbf{P}^1$ and why bases matter
Now switch \to $X=\mathbf{P}^1\times \mathbf{P}^1$. This surface has two natural rulings, and the Picard group has rank two:
- Let $F_1$ be the class of a fiber of the projection to the first factor.
- Let $F_2$ be the class of a fiber of the projection to the second factor.
Then
and the intersection numbers satisfy:
- $F_1\cdot F_1 = 0$ because two distinct fibers of the same ruling do not meet,
- $F_2\cdot F_2 = 0$ similarly,
- $F_1\cdot F_2 = 1$ because a fiber from each ruling meets in exactly one point.
A divisor class looks like $aF_1+bF_2$. If $D\sim aF_1+bF_2$ and $E\sim cF_1+dF_2$, then
This example is a lesson in how intersection theory encodes geometry:
- the two rulings create two independent directions of degree,
- intersection counts “cross terms,” not “self terms,” because fibers in the same direction do not meet.
Once you internalize this, you can compute intersections on many rational surfaces by choosing a good basis in $\mathrm{Pic}$.
Example 4: blowing up a point and the meaning of self-intersection
One of the first genuinely geometric operations in algebraic geometry is the blow-up. Let $\pi:\widetilde{\mathbf{P}^2}\to \mathbf{P}^2$ be the blow-up at a point $p$. The exceptional divisor $E$ is a copy of $\mathbf{P}^1$ sitting above $p$.
The Picard group becomes rank two:
where $H’=\pi^*(H)$ is the pullback of a line class.
The intersection form is determined by:
- $H’\cdot H' = 1$ (pullback preserves the line intersection away from the blown-up point),
- $H’\cdot E = 0$ (a general line avoids the exceptional divisor),
- $E\cdot E = -1$ (the exceptional curve has negative self-intersection).
That last number is not decorative. It encodes the fact that $E$ can be contracted back \to a point, and it is the first hint of how intersection theory interacts with birational geometry.
Proper transforms and how multiplicity changes
Suppose $C\subset \mathbf{P}^2$ is a curve of degree $m$ with multiplicity $r$ at $p$ (meaning $p$ is an $r$-fold point of $C$). Its proper transform $\widetilde{C}$ in the blow-up has class
Now compute the intersection of two proper transforms $\widetilde{C}$ and $\widetilde{D}$ of curves $C$ and $D$ of degrees $m,n$ with multiplicities $r,s$ at $p$:
Geometric meaning:
- $mn$ is the total intersection multiplicity in the plane,
- $rs$ is the contribution coming from the blown-up point,
- the blow-up removes that concentrated intersection and spreads it along $E$.
This is a concrete demonstration of how intersection theory manages singularities and base points. You do not “fix” a computation by wishing tangencies away; you change the space so the computation becomes clean.
Example 5: adjunction as an intersection computation
Intersection theory also organizes intrinsic invariants, like genus, through divisor classes. On a smooth surface $X$, the adjunction formula for a smooth curve $C\subset X$ says
where $K_X$ is the canonical divisor class.
On $\mathbf{P}^2$, $K_{\mathbf{P}^2}\sim -3H$. For a smooth plane curve $C\sim dH$, the formula gives
So
This is a remarkable compression:
- genus is a topological-looking invariant,
- it becomes a one-line intersection computation.
It also shows why the intersection pairing is not merely about counting points. It interacts with line bundles, differentials, and the global geometry of embeddings.
How the examples fit into the modern framework
The examples above can be reframed in the standard modern objects:
- Divisors correspond to line bundles via $D \mapsto \mathcal{O}_X(D)$.
- Intersection numbers can be interpreted using Chern classes:
– on a surface, $D\cdot E$ can be seen as $\int_X c_1(\mathcal{O}(D))\cup c_1(\mathcal{O}(E))$.
- On higher-dimensional varieties, intersection theory lives in the Chow ring $A^*(X)$, with products of cycle classes.
You do not need to master the full formalism to compute effectively. The habit that matters is the same one visible in the worked examples:
- translate geometry into classes,
- compute in the algebraic structure (Picard group, Chow ring),
- interpret the result.
A practical computation recipe you can reuse
When you face an intersection question in algebraic geometry, a reliable workflow is:
- Identify the ambient variety $X$ and compute or choose a basis for $\mathrm{Pic}(X)$ or $A^1(X)$.
- Express the subvarieties you care about as divisor classes in that basis.
- Use known intersection numbers on the basis elements to compute the desired product.
- If the situation involves singularities or base points, perform a blow-up and recompute using proper transforms.
- Translate the final number back into the geometric statement you actually care about.
Each step is a move you can justify with standard theorems, which is why intersection theory scales: it turns geometry into a controlled algebraic calculus.
Why intersection theory is a good thread for learning algebraic geometry
Intersection theory sits at a crossroads where many core themes meet:
- local algebra produces multiplicity,
- global geometry produces conservation laws like Bezout,
- line bundles and Picard groups package divisors,
- birational modifications like blow-ups change spaces but preserve computable invariants,
- adjunction links intersections to intrinsic invariants like genus.
If you can compute confidently in the examples above and explain what each computation is measuring, you have absorbed more than a set of facts. You have absorbed a style of reasoning that reappears everywhere in algebraic geometry: reduce to invariants that are stable under the operations the subject is built to perform, compute in a structure that behaves functorially, then reinterpret the result back in geometry.
Books by Drew Higgins
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