Yeah, it would probably have made more sense for me to try to explain what I know rather than what I don’t. Another thing that’s probably worth mentioning now that I think of it, when the article says these shapes are the ‘simplest’, I would assume they mean the algebraic equations that define them are the simplest in some sensible and well-defined fashion, and that means the shapes are simplest and most atomic in a way that would be too subtle for anyone to even notice otherwise.
No. In one (complex) dimension, the only Fano variety is CP^1 = S^2, the two-sphere. Locally, it looks like the complex plane, not a line in it. In 2D, you have an object with four real dimensions (or two complex dimensions). By the Whitney embedding theorem, it does have a smooth embedding in 8-dimensional real Euclidean space, but that ambient space really isn’t important here; it’s not intrinsic and is mostly irrelevant for this sort of thing. (For that matter, 8 is not necessarily the lowest possible dimension; an orientable, closed 2-manifold can live in R^3 without any problems.)
More concretely, the first Fano variety in (complex) dimension 2 is the complex projective plane CP^2. It’s given by triples [x; y; z] with some x, y, z nonzero, modulo the relation [ax; ay; az] = [x; y; z] for nonzero a. Equivalently, they’re complex lines through the origin of C^3. The other eight Fano varieties in dimension 2 are blowups of CP^2 at up to eight distinct points. At each of those points, take a small ball out of CP^2, take another copy X of CP^2 with the opposite orientation and another small ball out of it, and staple your original CP^2 and X together along the boundary while matching up the orientations. (In this case, an orientation is basically a choice of K or -K, where K is the canonical class of CP^2. The point is that you want to extend the canonical bundle of each part over the entire thing, and to do so you have to match them up at the boundaries. 4-manifold topology is very sensitive to choices of orientation in connected sums.)
Would Stranger on a Train be able to help at all?
I can’t imagine you are going to get a better explanation than Itself’s. I am out of my depth here although I know that “complete” implies that the field is algebraically closed (but it seems clear that here it is C) and the variety includes all points at infinity. The explanation of the 2 (complex) dimensional case was doubtless as clear as it is going to get. I cannot imagine why there is a limit of 8 attachments, however.
I think the important take-away for most of us is that these aren’t shapes like spheres or planes or cylinders. They’re defined mathematically over complex space in some way. And (I think…) those “shapes” in the images in the articles are all 3-D slices of some multidimensional thing.
I appreciate the attempt, Itself, thanks. I understand most of the words, and the image of stitching two CP^2s together, but actually understanding the whole thing isn’t going to happen.
You’re welcome; I just wish I could be more helpful. Unfortunately, algebraic geometry is a ridiculously esoteric subject, even by math standards, and there’s not much to explain it with besides pointing at a laundry list of definitions.