One of the basic things we can do in algebraic geometry is take a surface, take two curves lying on that surface, and ask how many points of intersection there are. In ‘intuitive’ situations, it is obvious how to do this, but it is also not hard to find situations where common sense might break down.

For example, if we take a line tangent to a conic in a projective plane over an algebraically closed field , does it intersect once or twice? Super-naive intuition would suggest ‘only once’, but if you think a little harder this answer seems kind of ridiculous, because for any other configuration of the line and the circle, the answer is ‘twice’, so by applying naive intuition we have created a disgusting discontinuity in our calculus of intersections.

Luckily, especially with the theory of schemes, this is easy to fix. If we have two curves intersecting at a (closed) point in some surface (everything smooth and projective just to be safe), look at the local function ring and you can write equations in this ring that locally describe the curves and . Thus in some sense the geometric space obtained by intersecting and at is best described as a scheme by which (assuming are distinct) is a local artinian ring over , hence of finite -dimension, and if you look at a couple of examples (e.g. the first question on Burt Totaro’s commutative algebra examples) it’s easy to convince yourself that this dimension is the correct notion of intersection multiplicity, denoted . So we have managed to justify a formal procedure for letting the answer to our original question be `2′, and thus remove our ugly discontinuity.

It turns out we can define a nice object that computes things like this for us. It is algebraically convenient to, instead of merely intersecting two irreducible curves, to intersect arbitrary -linear combinations of irreducible curves. Such a -linear combination is called a *divisor* and the group of all divisors on a surface is written .

In Hartshorne V,1 it is proved that there is a unique symmetric bilinear pairing satisfying the following two properties. Firstly, that if are smooth curves meeting transversally, then . The second property is a kind of continuity property which says that if two divisors are *linearly equivalent* then they behave identically with respect to the intersection pairing. It turns out that this pairing satisfies a more general version of property 1: that for any two curves with no common irreducible component,

In particular, notice how this pairing `agrees with us’ about the correct definition of intersection multiplicity.

You may well now be feeling fairly happy about what we have achieved, but there is one apparently silly but actually very important case we still have not come close to addressing. How many times does a curve intersect itself? The naive answer should be `infinitely often’, or thereabouts, but again think about how ugly this would be. If you take your curve and apply a small algebraic deformation, it will then intersect in some finite number of points. In fact, if there is a different curve in the same linear equivalence class, then our pairing is guaranteed to spit out a finite number (since it’s well-defined up to linear equivalence).

How do we explain what’s going on? In particular, is there a way of getting hold of these numbers and maybe finding a geometrical interpretation for them? To really get a grasp of this, we will need some of the tools of basic algebraic geometry (Hartshorne II and III). The plan is to continue working with and distinct curves (maybe even transversally intersecting) but to massage our definition of until it doesn’t look completely mad to consider .

Firstly, a divisor on a curve is just a formal -linear combination of points on the curve, and in fact in the situation we are studying there is a natural intersection divisor

This can be interpreted either as a divisor on or on and in each case is the degree of this divisor.

It is one of those cool facts in algebraic geometry that there is a surjective homomorphism , where is the group of invertible sheaves on (with tensor product as operation), and this map has kernel precisely equal to the principal divisors, so in particular it remembers what the degree of a divisor is. Thus instead of studying , we switch our focus to studying the corresponding invertible sheaf .

In fact, this game works on surfaces and their divisors too, so in particular an irreducible curve in gives rise to an invertible sheaf on , and such sheaves have the property that their duals are canonically isomorphic to the *ideal sheaf* of , defined by (where ). A huge advantage to this is that these sheaves have nice functoriality. In particular (returning to our intersecting curves ), if is the embedding of into , then we can pull back to an invertible sheaf on , and it is not difficult to prove exactness of:

From what we were just saying about ideal sheaves, this sequence says precisely that . In particular, we could accurately define the intersection number of two distinct irreducible curves and to just be the degree of (considered as a divisor on ), and our above argument shows that the apparent asymmetry of this formula is not a problem.

For our purposes, it also suggests the following possible answer to our question. Could it be that ? Miraculously, though it would be unthinkable to try pulling back the divisor on to a divisor on , we can always pull back a line bundle and get out another line bundle, so by switching from studying divisors to studying invertible sheaves, we really do get an element of that corresponds to intersected with itself, and indeed taking the degree of this is what the intersection pairing does.

Awesome. What does this look like? We’ll compute a pretty important example (which is the first step in the proof of the Riemann Hypothesis for curves over finite fields – something which I’ll probably say something about soon). Consider a smooth irreducible projective curve over an algebraically closed field , and let , a surface. Let be the embedding of in the diagonal of . We shall compute the intersection number , and note that it’s pretty interesting.

The main result of this post convinces us that our task is to calculate the degree of . But is the ideal sheaf of , that is to say the kernel of , so as -modules,

But we know what this sheaf is. Recall (Hartshorne II,8) that the sheaf of differential forms is defined precisely as , which, since we are in dimension 1, is also the canonical sheaf. Thus our sheaf is precisely the dual of the canonical sheaf , and in particular, it has degree . So the intersection of a curve with itself recovers the Euler characteristic.

This post has been largely improvised by me while trying to simultaneously learn some basic algebraic geometry and figure out how to prove the Weil conjectures, so is bound to contain some errors or things I could have done more clearly, which readers are encouraged to point out. The obvious reference is Hartshorne V. I may be back very soon to prove the Riemann hypothesis for curves over a finite field, which essentially boils down to computing certain intersection numbers between divisors in , and then applying the Hodge index theorem, which places constraints on how the intersection pairing may behave and lets you write down an inequality which can be massaged into the Hasse-Weil inequality, which essentially is a tight version of the `prime number theorem’ for curves over finite fields that is tight enough to imply the Riemann hypothesis.

Also, this surely has connections to the algebraic topological theory surrounding the Lefschetz fixed-point theorem, in a way I haven’t properly thought about yet, so if anyone knows any interesting things about that I’d enjoy talking about it, and I vaguely recall somebody telling me that intersecting curves with themselves becomes clearer in a (Lurie style) derived algebraic geometry picture. Again, I know nothing and would be interested to know something. Thanks.

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January 31, 2012 at 11:57 pm

Zhen LinYou probably know this already, but if not, here is the connection with the Lefschetz fixed point formula. Consider the action of the Frobenius p-power map on (the separable closure of) F_q. Its fixed points are precisely the elements of F_p. Now suppose we have a smooth algebraic variety V over F_p, and we want to count the F_p-rational points of V… wouldn’t it be nice if we had a formula which told us how many fixed points an endomorphism of V has? Hartshorne describes this in Appendix C.

February 1, 2012 at 8:42 am

tloveringI was thinking more about the fact that the Lefschetz number of the identity map is the Euler characteristic, which in this post we’ve shown to be equal to the intersection number of a curve with itself.

March 11, 2013 at 5:26 am

Dmitry VaintrobI was also thinking about this recently, and I think that as long as you’re just intersecting curves on surfaces, the “Lurie-style” explanation is pretty easy and significantly predates Lurie. Namely in this case the intersection factors through K theory, i.e. the skyscraper functor “Subvarieties” -> “Sheaves”. Intersection corresponds to tensor product. If your intersection is zero-dimensional (and maybe satisfies some mild niceness condition) then the straight-up tensor product is a sum of skyscrapers supported at points, and the intersection multiplicity is the dimension of global sections. As things start getting worse, you have to derive both tensor product and global sections, and replace dimension by Euler characteristic – but I think that’s all you have to do, even in characteristic p (as long as things are smooth). Actually, I’ve always wondered why people put so much effort into various theories of “cycles” when much of the time thinking about sheaves is enough. I suppose this has something to do with Hodge theory or something — maybe you know of a good explanation.