Heisuke Hironaka Has Died

You should be fine if you have an undergrad degree in math, engineering, physics, or some other hard science and have taken a course in advanced calculus/baby real analysis -- whatever it is that they call "calculus with proofs" nowadays. They try to hide all the algebraic geometry by talking about zero sets of systems of polynomial equations. Whether you are talking about "an algebraic variety", "manifold", curve, surface, etc, it's just the zero set of polynomials of possibly one, two, three dimensions. The fact that it is the zero set of a polynomial equation means you have nice properties -- that is what algebraic geometry is about -- studying these zero sets. And this paper tries to resolve singularities in the zero sets.

That said, papers like these have to be read slowly and carefully. If you need some help with understanding what this is about, the basic idea is that you want everything to be non-singular - no vanishing derivative, no undefined derivative, no self-crossing, etc. Locally, every one-dimensional curve should be a flat line, every two dimensional surface a plane, etc. Most of the theorems apply to non-singular varieties.

In the real world, things are singular at "bad" places, and you have to deal with that. So what you do is try to find a regularization, or a variety that is close to the one that you want to study, but with the singular parts smoothed out, and then a nice map from the smooth variety to the singular variety that hopefully does not change too much so that you can prove your result for the singular variety, and then see how it changes under your map.

In practice, these maps are called blowups and blow-downs. E.g. if you take a curve that crosses back on itself in the plane, then embed the curve into three dimensional space and pull it apart so in the three dimensional space it does not cross itself. Then the projection back onto the plane is your map from the non-singular to the singular curve. If you can always do this (and you can) then you can think of singular curves as "shadows of non-singular curves"

Algebraically, you can think of it this way:

take a nodal cubic that crosses itself at the origin:

```y^2 = x^2(x+1)``` This has a singular point. We want to make a new curve without this singular point, and a nice projection map from the new curve to the old. We do this with two charts.

The first chart is when x is not zero. ``` chart 1: y = ux Then we have u^2x^2 = x^2(x+1) and we can factor out x^2 to get u^2 = x+1 in the (u,x) plane and the special solution x^2 = 0.

```

The second chart is when y is not zero ``` chart 2: x = vy Then we have 1- v^2 - v^2 = 0 in the (y, v) plane and the special solution y^2 = 0 ```

The two charts can be glued together nicely in the area where they are both defined by the transforms x = vy, y = ux (or v -> 1/u, x ->y), so the larger curve lives in the union of both charts. These charts and their transitions are projective space as they represent the set of lines in the plane, so a blow up is always done in projective space (this generates to multiple variables easily).

Now this larger curve reduces to the original curve at the point u = 1 in one chart and v = 1 in the other chart. Our blow down map consists of substituting u = 1 or v = 1).

The special case solutions x^2 = 0 in chart 1 and y^2 = 0 in chart two are the "exceptional divisor" or a portion of the non-singular curve that is codimension one and is compressed to zero dimensions by the blow-down map. Geometrically, when you project from three dimensions to two dimensions you flatten one dimension, that flattened dimension is the exceptional divisor. It also defines the map, because the flattened dimension determines the projection.

The exceptional divisor is exactly the location of points where your blow down map is not "smooth".

But one blow up at a point usually isn't enough. If you have higher order zeroes, you may need multiple blow ups, and you can have many degenerate points and need to blow up many times. So the idea is to keep blowing things up until you get something smooth. And then you compose all your previous blow up maps to get your blow down map from your smooth variety back to your singular variety. This was all known in the 19th Century.

Where this approach may fail is that it's possible to blow something up but not have the result be smoothed out, and this happens when the exceptional divisors do not have smooth crossings. So Hironaka needed to prove that there always exists exceptional divisors with smooth crossings, which is why Hironaka needed to track all the other divisors and use dimensionality arguments to show that there exists a transverse divisor to all of them. Additionally, he needed to find smooth singular surfaces to blow up, which is where his "hypersurface of maximal contact" approach fits in, and this is done via induction on the dimension. These are the primary obstacles Hironaka overcame and are covered in detail in the paper.