The purpose of this post and the next is to work through a basic example of intersection theory: intersections of curves on a surface. This is a fundamental and basic example in algebraic geometry, and since I’ve never studied intersection theory, it like seems a reasonable place to start. The references here are chapter 5 of Hartshorne’s Algebraic geometry and Mumford’s Lectures on curves on an algebraic surface.

1. Curves on surfaces

The subject of “curves on a surface” is the subject of Mumford’s book mentioned above; the purpose of this section is simply to set down the definitions.

Let {k} be an algebraically closed field. A surface {S} is a smooth projective surface over {k}. There is a classification of surfaces, but let’s just list a couple of basic examples: {\mathbb{P}^2, \mathbb{P}^1 \times \mathbb{P}^1}, (smooth) hypersurfaces in {\mathbb{P}^3}, and ruled surfaces.

Definition 1 curve on a surface {S} is an (effective) divisor on {S}. Equivalently, it is a subscheme {C \subset S} pure of codimension one, so locally cut out by one equation. (But {C} is not necessarily smooth, or even reduced.)

The goal of this post and the next is to set up a basic intersection theory for curves on surfaces. Given two curves {C, D \subset S}, we’d like to define the intersection product {C.D}. There is one case where it is easy: suppose {C} and {D} meet only transversely. In other words, for each {p \in C \cap D}, we choose local equations {f,g \in \mathfrak{m}_{S, p} \subset\mathcal{O}_{S, p}} for the subschemes {C, D}, and

\displaystyle (f,g) = \mathfrak{m}_{S, p}.

In particular, this implies that {C, D} are nonsingular at all points of intersection. In this case, we would like to require

\displaystyle C.D = \sum_{p \in C \cap D} 1 \quad (\text{if transverse intersection}). \ \ \ \ \ (1)

Once we require the above condition and two more natural conditions, we will prove that the intersection product is uniquely determined:

  • The equation (1) holds under transversality assumptions and if {C, D} are smooth.
  • The intersection product is additive. That is, given curves {C_1, C_2, D}, we have

    \displaystyle (C_1 + C_2). D = C_1.D + C_2.D,

    where {C_1+C_2} is treated as an effective Cartier divisor.

  • The intersection product is invariant under linear equivalence. If {C, C'} are linearly equivalent curves, we want

    \displaystyle C. D = C'.D,

    so that the intersection product is invariant under deformation. In particular, this and the previous item show that the intersection product only depends on the line bundle associated to a divisor (and can make sense for any divisor, not necessarily effective).

Our goal is to prove:

Theorem 2 There is a unique pairing

\displaystyle \mathrm{Pic}(S) \times \mathrm{Pic}(S) \rightarrow \mathbb{Z}

satisfying the above three conditions.

2. Existence of the intersection pairing: complex case

Let {M} be a smooth {n}-dimensional oriented manifold. Given oriented submanifolds {N \subset M, N' \subset M} of complementary dimensions intersecting transversely, we can compute the size of the intersection {N \cap N'} using “calculus” (or cohomology): let {[N], [N'] \in H^*(M; \mathbb{C})} be the fundamental classes. Then we have:

\displaystyle | N \cap N'| = \int_M [N] \cap [N'],

where “integration” means actual integration in the de Rham theory, or equivalently pairing with the fundamental homology class.

Caveat: The “cardinality” of {N \cap N'} is really a signed one: each point is counted with an orientation. In the algebraic (or complex) case, this is not a problem: the local intersection multiplicities for transverse intersections are always positive as the orientation on a complex vector space is canonical.

Let’s now imagine that {S} is a complex algebraic surface. In this case, given a curve {C \subset S}, we can associate to {C} a fundamental cohomology class {[C] \in H^*(S; \mathbb{C})} (even when {C} is singular!); this is equivalently the first Chern class of the (holomorphic) line bundle {\mathcal{O}(C)} on {S}. Alternatively, it is the form defined by the property

\displaystyle \int_S \omega \wedge [C] = \int_C \omega|_C,

for all forms on {C}. (Note that the integral vanishes unless {\omega} is a {(1,1)}-form, so that {[C]} can itself be taken to be a {(1,1)}-form.)

In the Hodge decomposition for {H^*(S; \mathbb{C})}, which yields

\displaystyle H^2(S; \mathbb{C}) \simeq H^{0,2}(S; \mathbb{C}) \oplus H^{1,1}(S; \mathbb{C}) \oplus H^{2, 0}(S; \mathbb{C})

the fundamental classes {[C]} therefore live in {H^{1,1}(S; \mathbb{C})}. The Lefschetz theorem states that such fundamental classes {[C]} span the {(1,1)}-cohomology. (The generalization of this to higher dimensions is the famous Hodge conjecture.)

Here is a possible definition of the intersection pairing in the complex case:

Possible definition: Given curves {C, C' \subset S}, the intersection product is

\displaystyle (C.C') {=} \int_S c_1(\mathcal{O}(C)) \wedge c_1(\mathcal{O}(C')) = \int_S [C]\wedge [C'].

Intersection theory in differential topology, as quoted above, implies that this gives the correct answer when {C, C'} are smooth curves intersecting transversely. The rest of the properties are evident; the fact that the Chern classes live in integral cohomology implies that the intersection product actually takes values in {\mathbb{Z}}. So, at least over the complex numbers, this gives a satisfactory definition of the intersection pairing.

3. Uniqueness of the intersection pairing

The above result, constructing the intersection pairing over {\mathbb{C}}, is not really satisfactory if we are working in characteristic {p}. (We could try to use étale cohomology to replace complex cohomology, but we certainly do not need such heavy machinery… .)

Let’s first start by checking uniqueness of any intersection theory. In other words, the intersection product {(C.D)} is determined by what happens in the “nice” case of smooth curves with transverse intersection. We can’t generally deform a pair of curves to this case (in fact, this impossibility leads to negative intersection products), but we’ll see that we can after adding a sufficiently ample divisor.

Choose an ample divisor {H} on the surface {S} (say, the intersection of the surface with a hyperplane in some projective imbebdding). Then for {n \gg 0},

\displaystyle nH, n H + C, n H + D

are all very ample line bundles. We now need to invoke a theorem of Bertini about the generic hyperplane section of a smooth variety {X} imbedded in projective space:

  • The generic hyperplane section is smooth.
  • (If {\dim X \geq 2}.) The generic hyperplane section is connected. This isn’t really necessary for us.

In view of this, choose a generic divisor linearly equivalent to {nH + C}: by Bertini’s theorem, it can be represented by a smooth, connected curve {C_1} on the surface {S}. The generic divisor linearly equivalent to {nH + D} is a smooth connected curve {C_2} on {S}, which moreover intersects {C_1} in a smooth scheme (thus transversely). Similarly, the generic divisor linearly equivalent to {nH} is a smooth, connected curve {C_3} on {S}, which intersects {C_1, C_2} transversely.

In other words, by choosing {n \gg 0} and appropriate divisorial representatives, we are in the “nice” situation for {nH, nH + C, nH + D}. The pairwise intersections here are already determined by equation (1); this forces {C.D}.

4. Existence (incomplete)

But now we need to check existence. The above proof suggests a way of defining the intersection pairing through repeated use of Bertini’s theorem and hyperplane sections. However, it’s also possible to give a direct, computationally useful formula. We’ll start towards that formula in this section.

Let’s go back to the “nice” situation where it was clear what the intersection number should be: this was where two curves {C, D} were nonsingular and intersected transversely. (In fact, we just needed to assume that they intersected transversely: they could be singular elsewhere.) Why is this case so nice? One reason is that the intersection

\displaystyle C \cap D = C \times_{S} D

is a “homotopy invariant” intersection, which moreover consists of just a direct sum of reduced points. In other words, we have

\displaystyle k \simeq \mathcal{O}_{C, p} \otimes_{\mathcal{O}_{S, p}} \mathcal{O}_{D, p} \simeq \mathcal{O}_{C \cap D, p}, \quad p \in C \cap D

by definition, but in the “nice” case the tensor product also happened to be the (homotopy invariant) derived tensor product. If {R} is a ring and {(x,y)} is a regular sequence in {R}, then

\displaystyle R/(x) \stackrel{\mathbb{L}}{\otimes} R/(y) \simeq R/(x,y).

In general, without this regularity assumption, this completely fails.

This suggests that the problem with defining naively

\displaystyle C \cap D = |C \cap D|

is two-fold:

  • The scheme { C \cap D} may be non-reduced: there may be “fuzz” on the points in it that should be taken into account when computing intersection multiplicities.
  • The “homotopy-theoretic” tensor product {\mathcal{O}_C \stackrel{\mathbb{L}}{\otimes}_{\mathcal{O}_S} \mathcal{O}_D} might not be the same as {\mathcal{O}_{C \cap D}}. There might be higher homotopy groups in {\mathcal{O}_C \stackrel{\mathbb{L}}{\otimes}_{\mathcal{O}_S} \mathcal{O}_D}, which can be thought of as a higher version of nilpotents, and which also should be accounted.

The first problem is a problem with working with classical varieties, which don’t see nilpotents, and which the use of schemes allows us to solve. The second problem can’t be solved only in the framework of schemes, but one of the motivations of derived algebraic geometry is to form a “homotopy-theoretic” fiber product

\displaystyle C \stackrel{h}{\times}_{S} D

whose structure sheaf is in fact the derived tensor product {\mathcal{O}_C \stackrel{\mathbb{L}}{\otimes}_{\mathcal{O}_S} \mathcal{O}_D} (although the structure sheaf is no longer a sheaf of ordinary commutative rings!).

Let’s work in a thought bubble, and assume without comment that this is perfectly meaningful. In this case, we might imagine that we can solve our problems by defining

\displaystyle C.D = \mathrm{size} ( C \stackrel{h}{\times}_S D),

where we have to specify what “size” means. The Euler characteristic of the structure sheaf (which we’d want to be { \mathcal{O}_C \stackrel{\mathbb{L}}{\otimes}_{\mathcal{O}_S} \mathcal{O}_D}) is a decent invariant. We might thus define:

Definition 3 We define the intersection multiplicity

\displaystyle C.D = \chi( \mathcal{O}_C \stackrel{\mathbb{L}}{\otimes}_{\mathcal{O}_S} \mathcal{O}_D );

more generally this allows us to define the intersection multiplicity of line bundles.

The Euler characteristic is the alternating sum of the dimensions of the (hyper)cohomology groups.

This definition is, in fact, no different from the one without the derived tensor product unless {C, D} share a common irreducible component: the higher {\mathrm{Tor}} groups will vanish. In higher dimensions, this higher terms are necessary, and this is Serre’s intersection formula. Nonetheless, this point of view will prove useful in motivating a more explicit version of the above formula in the next post.

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