Continuing my departure from my usual narrative, I’m going to talk about Resolution of Singularities. This is a very classical topic, and research continues in it to this day. I’ve been interested in singularities for awhile and I’m currently reading Lectures on Resolution of Singularities by Kollár, which is very accessible and includes a full proof, which he admits is really, like all other proofs of it, a modernization of Hironaka’s 1964 tour-de-force.

So what IS a resolution of singularities? Well, we start with an arbitrary quasiprojective variety V. Then the weakest version is the claim that there exists another variety V' and a map \pi:V'\to V such that V' is smooth, such that V'\to V is a birational morphism (that is, it has an inverse defined on the complement of finitely many subvarieties of V) which is projective. All that being projective means is that the map factors as V'\to \mathbb{P}^n\times V\to V where the first map is a closed immersion and the second map is the projection.

Now, even this weakest version is still open in positive characteristic, so we’re going to specialize to complex varieties, which means that smoothness is the same as being a complex manifold.

We’re going to avoid most of the technical details, except for statements and explanations (and maybe a sketch of special cases that are easier), and instead focus on the history of Resolution and on where it fits into mathematics, as well as sketching a few techniques.

So first we’ll start with complex curves. The earliest result that can reasonably called a resolution of singularities result is actually due to Newton in 1676. It was very analytic, and was specialized to algebraic plane curves, and generally isn’t the best method, but it was a start. However, the modern method of describing the resolution of curve singularities is a little bit more technical, but rather nifty.

We’ll describe the basic phenomenon more generally first, though. Take a variety V, it can be covered by affine varieties, which are the same things as nice rings. However, there’s a way to make the nice rings nicer. For each open set, remember how they’re glued together, and then normalize, that is, replace the rings with their integral closures in their fields of fractions. Then use the same gluing maps (ok, not EXACTLY the same, but you get the new from the old) to put it back together. This gives a new variety \tilde{V} called the normalization of V. A variety is called normal if it is equal to its normalization, and normalizing twice has no extra effect. So there’s a theorem which says that normal varieties are smooth in codimension one. That is, there aren’t any “big” singularities. For instance, on a surface, there are no singular curves, just isolated points.

For curves, this means that the normalization IS the resolution! This is because there’s nowhere for the singularities to go except codimension one, and then normalization kills them. Normalization is very useful for resolution of surfaces as well, which we’ll get to in a minute, once we’ve discussed two more tricks for resolving curves (and they all give the same answer, which is good).

The second trick for resolving curves that we’re going to talk about is resolution by blowing up points. More generally, you can blow up subvarieties of a variety. What it does, in effect, is that it stretches out the subvariety into something of codimension one. If you started with something of codimension one, it picks it apart a bit.

As an example, take a curve that intersects itself at a single point. Blowing it up at that point pulls the two branches apart and we get something with no self-intersections out. From this example, it’s not too hard to believe that by blowing up the singular points of a curve and repeating as necessary, after finitely many steps this gives a resolution.

The third method is really clever, and is due to Albanese in 1924, so it came late in the game for curve singularities, but it is how surfaces and threefolds were first resolved. It starts by taking a projective variety, which is a shame, because not all varieties fit into any projective space. Then it uses projections from singular points onto smaller projective spaces. This method actually doesn’t resolve all singularities, but what it does is that it eventually leads to a variety with no points that are worse in some numerical way to the dimension factorial. So for a curve, this is 1, which means resolution. For surfaces it said 2, so only double points are left to work with, and those can be handled. This is really what Albanese was after back in 1924, resolution for surfaces, though he didn’t make a completely rigorous argument.

Complex surfaces were finally resolved by Walker in 1935, and the result was extended by Zariski to all characteristic zero fields in 1939. Zariski’s student, Abhyankar, was able to extend the result to positive characteristic in 1956.

In 1944, Zariski was finally able to bring about a resolution of 3folds over the complex numbers, and then in 1966, Abhyankar got it to work for 3folds in positive characteristic, at least for characteristic greater than 5.

Zariski’s proof for 3folds was the end of the story for twenty years until another student of his, Heisuke Hironaka in 1964 published his landmark papers Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero I and II.

Hironaka actually got a much stronger result than the weak theorem mentioned at the beginning. His result is the following:

Theorem: Let k be a field of characteristic zero (such as \mathbb{C}) and let X be a variety over k. There exists a subscheme of X (more or less, we count things with multiplicities, so we can have twice a point, or three times a curve) which consists of precisely the singular points of X and such that blowing up D gives a smooth variety X' such that \pi:X'\to X is a birational morphism.

This immediately gives that away from the singular points, \pi is an isomorphism.

Another way to phrase this involves instead of blowing up the singular points “with multiplicity,” we can instead perform a finite sequence of blowups of singular points.

Now, this put the question of whether resolutions exist at all firmly to rest, a problem which had been plaguing algebraic geometry since at least 1899, when Levi first attempted to resolve surfaces (unsuccessfully). However, resolution wasn’t done. People wanted stronger versions of the result, and it’s still not known in positive characteristic. Hironaka’s proof was simplified and made more powerful by many people, and one of the stronger forms is that, given X a variety in characteristic zero, there exists a variety X' and a projective map X'\to X such that X' is smooth, f is birational and an isomorphism on the nonsingular points, and that the inverse image of the singular points is what is called a simple normal crossing divisor. All this is is that it is a union of hypersurfaces, each of which is smooth, which intersect transversely, that is, in the simplest possible way.

There are even stronger resolution theorems for characteristic zero now, but they get a lot more technical very quickly, and a lot of the geometric intuition is left behind.

So now that we know this, what does that get us? Well, there are a lot of strong results that only hold for smooth varieties that we want to be able to extend to arbitrary ones, and projective birational maps like the ones promised by resolution are very helpful in doing so. What it, in essence, allows is for us to in many cases pull a problem back from a singular variety to a smooth one, solve it there, and then push the solution forward and see what we get.

One example is Deligne’s work on Hodge Theory, which takes the cohomology of singular varieties and expresses it in terms of the cohomology of a resolution. He literally references Hironaka’s paper in the third paragraph of his paper Théorie de Hodge, II. Hodge theory is still a very active field of research in algebraic geometry, and it gives rise to the Hodge Conjecture, which is one of the millenium problems…and which no one has a clue how to do in general, but which Deligne has done a lot to demonstrate that it is likely to be true (by independently proving consequences of it).

Aside from the final results of the quest for resolutions, the work increased greatly out knowledge of how specific singularities behave. This is important for dealing with specific examples and seeing how things work. In fact, canonical singularities (which I will not define) cannot be defined without a resolution. For instance, it comes up the Minimal Model (or Mori) Program which is attempting to solve the bigger and older problem of birational classification of algebraic varieties. That is, when are two varieties birational, and how many birational classes are there? The latter question generally has to be answered by a moduli space, because there aren’t even finitely many birational classes for curves, and higher dimensions are worse.

Resolution is also useful for generalizing Kodaira vanishing, which says that for X a projective smooth variety and L an ample line bundle then for i>0 we have H^i(X,\omega_X\otimes L)=0, where \omega_X is the canonical bundle. Now a theorem of Grauert and Riemenschneider says that for an appropriately defined K_X, we have the result for L semiample and big on any variety, where semiample means that some power of it is generated by global sections and big means that \dim H^0(X,L^m)>cm^{\dim X} for large enough m and some positive c. Just how is this mysterious K_X defined, though? Well, you push the canonical bundle forward from a resolution. This theorem is apparently used for a lot of things in the Mori program as a standard vanishing theorem for cohomology.

Overall, the work on resolution has been and continues to be a central area on algebraic geometry, and Hironaka’s paper gave the first correct proof of this important result. For more detail on who did what and when, see Resolution of Singularities. The first part is available on Google Books.