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Stream: learning: questions

Topic: punctual Hilbert schemes

John Baez (Jun 07 2021 at 00:02):

Has anyone here thought about punctual Hilbert schemes?

Morgan Rogers (he/him) (Jun 07 2021 at 09:22):

What's one of those?

Tim Hosgood (Jun 07 2021 at 12:35):

as in, Hilbert schemes parametrising discrete sets of points?

Tim Hosgood (Jun 07 2021 at 12:35):

if so, then yes, a bit, thinking of them as resolutions of configuration spaces

John Baez (Jun 07 2021 at 16:03):

Yes, Hilbert schemes describing sets of points and how they collide: the nth punctual Hilbert scheme of a commutative algebra A, for example, has as its closed points all the ideals J of A such that dim(A/J) = n.

John Baez (Jun 07 2021 at 16:07):

Do you know anything about how punctual Hilbert schemes are related to configuration spaces, @Tim Hosgood? That's actually one of the things I'm wondering about. The configuration space whose points are n-tuples of distinct points in an affine variety X seems to inject into the nth punctual Hilbert scheme of the corresponding algebra.

John Baez (Jun 07 2021 at 16:09):

When I hear "configuration space" I think of a space whose points are (ordered or unordered) n-tuples of points in some space X.

John Baez (Jun 07 2021 at 16:11):

If I want to let the points "collide" I think of the Chow variety of X formed by taking $X^n$ and modding out by the action of $S_n$.

John Baez (Jun 07 2021 at 16:13):

There's a map from the punctual Hilbert scheme to the Chow variety, I've heard, and that makes plenty of sense.

Tim Hosgood (Jun 07 2021 at 16:38):

yes, the Hilbert scheme $\operatorname{Hilb}^n(X)$ comes with a map (called the Hilbert–Chow morphism) into the Chow space $X^n/S_n$, and the Hilbert scheme is nice (i.e. an irreducible smooth variety), and can thus be thought of as a resolution of the Chow space

Tim Hosgood (Jun 07 2021 at 16:41):

if you take a nice example like the affine plane $X=\mathbb{A}^2$, then you can understand the Hilbert–Chow morphism pretty concretely: the Hilbert scheme is covered by open affines $U_\lambda$ consisting of ideals of $I\triangleleft\mathbb{C}[x,y]$ such that $\mathbb{C}[x,y]/I$ has a certain basis, where $\lambda$ corresponds to a partition of $\{1,\ldots,n\}$ (and this is where Young diagrams come in), and the Hilbert–Chow morphism is then given by sending an ideal $I$ to its support $\operatorname{Supp}(I):=\mathbb{C}[x,y]/I$ (i.e. the subset of points where all polynomials in the ideal vanish)

Tim Hosgood (Jun 07 2021 at 16:45):

generally what people refer to as the punctual Hilbert scheme (at least when $X$ is an affine space) is the smaller thing that consists of ideals supported at the origin, since this tells you about what you mentioned: how points "collide"

Tim Hosgood (Jun 07 2021 at 16:46):

http://people.math.harvard.edu/~bejleri/Hilbertschemes_gss.pdf is a really nice introduction to this stuff

John Baez (Jun 07 2021 at 18:56):

Thanks! I found Bertin's introduction to punctual Hilbert schemes to be incredibly bureaucratic and unenlightening - it reads more like an introduction to the foundations of algebraic geometry (all the stuff I've never really wanted to learn).

John Baez (Jun 07 2021 at 18:57):

This looks better.

John Baez (Jun 07 2021 at 18:58):

I just found out that for varieties of dimension 3 or more, there are points in the punctual Hilbert scheme that aren't "limits of colliding points". :scared:

John Baez (Jun 07 2021 at 18:58):

https://twitter.com/gro_tsen/status/1401961668687564800

@johncarlosbaez Actually, if X is :eight_spoked_asterisk:︎any:eight_spoked_asterisk:︎ smooth projective variety of dimension ≥3, then for n≫0, the Hilbert scheme of 0-dimensional subschemes of length n of X is reducible (so it has other components than the “obvious” one consisting of specializations of n distinct points).

- Gro-Tsen (@gro_tsen)

John Baez (Jun 07 2021 at 18:59):

https://twitter.com/gro_tsen/status/1401962021294329857

@johncarlosbaez Ref.: Iarrobino, “Reducibility of the families of 0-dimensional schemes on a variety”, ‘Inv. Math.’, 15 (1972), 72–77. https://link.springer.com/article/10.1007/BF01418644

- Gro-Tsen (@gro_tsen)

Tim Hosgood (Jun 07 2021 at 19:07):

oh, that's an unnerving result indeed!

Tim Hosgood (Jun 07 2021 at 19:09):

what's really weird though is if you look at the (weak) bounds given in the paper: it proves that, in three dimensions, $\mathrm{Hilb}^n$ is reducible for $n>102$, which is quite a lot of points!

Tim Hosgood (Jun 07 2021 at 19:09):

but I also have absolutely zero intuition for why 103 points should suddenly be somehow worse than 102

Tim Hosgood (Jun 07 2021 at 19:10):

although maybe the bounds are much lower for e.g. the affine plane

Tim Hosgood (Jun 07 2021 at 19:16):

but for the affine plane, the Hilbert scheme is always irreducible, thankfully (https://mathoverflow.net/a/20707/73622)

John Baez (Jun 08 2021 at 05:44):

Does that paper prove that in 3 dimensions $\mathrm{Hilb}^n$ is reducible for n = 102?

I doubt it. I doubt that bound is sharp.

John Baez (Jun 08 2021 at 05:54):

Hmm, actually that MathOverflow article you cited points to another paper written 46 years later that seems to show that in 4 dimensions $\mathrm{Hilb}^n$ is reducible for $n \ge 8$. And this paper seems to give a constructive proof, rather than a dimension-counting argument.

Tim Hosgood (Jun 08 2021 at 11:33):

oh, i’d really like to understand a constructive proof, too see what these other components look like. i’ll give it a read

John Baez (Jun 08 2021 at 15:37):

If you make any progress, even a tiny bit, please let me know. I sort of bounced off this paper.

Tim Hosgood (Jun 08 2021 at 23:40):

Screenshot-2021-06-09-at-00.39.22.png Screenshot-2021-06-09-at-00.39.15.png Screenshot-2021-06-09-at-00.39.00.png

Tim Hosgood (Jun 08 2021 at 23:40):

looking at these pictures, it doesn't surprise me that you're interested in these things! very intriguing indeed

John Baez (Jun 09 2021 at 00:07):

Where did you see those pictures?

My real reason for being interested in these things is trying to get a clear visual understanding the geometric McKay correspondence, which gives a way to get the E8 Dynkin diagram out of the double cover of the symmetry group of the icosahedron:

https://johncarlosbaez.wordpress.com/2017/07/02/the-geometric-mckay-correspondence-part-2/

John Baez (Jun 09 2021 at 00:08):

But this correspondence proceeds via an "equivariant Hilbert scheme of points", so I need to understand that better.

Tim Hosgood (Jun 09 2021 at 00:32):

John Baez said:

Where did you see those pictures?

https://eudml.org/doc/142485

John Baez (Jun 09 2021 at 03:15):

My browser will not let me go to that site - it thinks it's dangerous.

Nathanael Arkor (Jun 09 2021 at 07:42):

Forbidden knowledge.

Tim Hosgood (Jun 09 2021 at 16:11):

Briancon, Joel. "Description de $\mathrm{Hilb}^n\mathbf{C}\{x,y\}$" Inventiones mathematicae 41 (1977), 45–90.

Tim Hosgood (Jun 09 2021 at 16:12):

John Baez said:

If you make any progress, even a tiny bit, please let me know. I sort of bounced off this paper.

so it seems like the bit that is of interest to you is section 5, where they explicitly construct a point $J$ in the non-smoothable component

John Baez (Jun 09 2021 at 16:33):

Thanks, I'll check that out.

Tim Hosgood (Jun 09 2021 at 16:35):

i must admit, i'm surprised that i've yet to read an explanation in the form that i think you want (i.e. "most of the points correspond to particle collisions, except for these exceptional points, which correspond to ???"

Tim Hosgood (Jun 09 2021 at 16:35):

if this is still a mystery then maybe i'll ask some of my algebraic geometry colleagues

John Baez (Jun 09 2021 at 16:38):

Right, that's the sort of explanation I want. Maybe nobody knows a simple explanation, which then makes this a nice problem... though not one I'm likely to solve!

John Baez (Jun 09 2021 at 16:39):

So yes, if you know people who are good at this sort of thing, please ask them.

Tim Hosgood (Jun 10 2021 at 12:52):

Screenshot-2021-06-10-at-13.52.21.png
(from https://arxiv.org/pdf/0812.3342.pdf)

John Baez (Jun 10 2021 at 20:42):

Nice!

Simon Burton (Jun 11 2021 at 11:51):

I thought this discussion of punctual Hilbert schemes was about Allen Knutson's comments on your Plethysm paper: https://golem.ph.utexas.edu/category/2021/06/schur_functors_and_categorifie.html#c059823

John Baez (Jun 12 2021 at 01:29):

That's probably why I got back to thinking about punctual Hilbert schemes. I'd love to understand what he's talking about. But my real goal here is to finish up my project of understanding the geometric McKay correspondence in simple terms.

The crowning glory of this story should go like this: The 120-cell is a 4-dimensional regular polytope with 120 faces, each shaped like a regular dodecahedron.

The space $X$ of all ways of putting a 120-cell in $\mathbb{C}^2$, centered at the origin, of any radius including zero, is a singular 2-dimensional complex variety. It has a singularity corresponding to the 120-cell of radius zero. We can resolve this singularity and get a smooth complex variety $\hat{X}$. The map $\hat{X} \to X$ is 1-1 except that a bunch of points map to the 120-cell of radius zero. These points lie in 8 spheres, which intersect each other in this pattern:

E8 Dynkin diagram

John Baez (Jun 12 2021 at 01:31):

This is amazing, and I've never seen a simple explanation.

Tim Hosgood (Jun 12 2021 at 03:16):

where can i read a non-simple explanation?

Simon Burton (Jun 12 2021 at 10:07):

I found this masters thesis by Max Lindh to be very helpful: http://uu.diva-portal.org/smash/record.jsf?pid=diva2%3A1184051&dswid=5245