Timeline for Philosophical underpinnings of Grothendieck's construction of the Hilbert scheme
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| Sep 13, 2020 at 1:11 | comment | added | Tabes Bridges | A couple of random thoughts: the Chow variety, which parametrizes cycles, seems to go hand in hand with the primitive language of varieties, with its destructive operations like taking the radical of an ideal. There is a great deal of information lost, and if I recall correctly, the Chow variety is not functorial. The cleaner notion of a scheme made a cleaner construction possible. As was often the case though, I have a feeling Grothendieck was the only person both bold and blind enough (in terms of his minimal background in classical AG) to think that such a thing was possible. | |
| Sep 12, 2020 at 10:30 | comment | added | Balazs | Historically the Chow variety came first, and before that, the Grassmanian. All these are attempts to parametrise classes of sub-objects of projective space. I would say the new "philosophical" underpinning of Grothendieck is the functor of points point of view, and the related notion of a flat family. The results you mention are tools to make this work. | |
| Sep 12, 2020 at 2:10 | comment | added | Yellow Pig | Sorry, I suspected from the beginning that my comment about Hilbert didn't have much depth (and I am certainly aware of the fact that Hilbert invented the Hilbert polynomial :) ), but I guess I'm still curious, did the idea of Hilbert schemes first occur to Grothendieck, or were there precursors to it? | |
| Sep 11, 2020 at 23:57 | comment | added | ssx | @YellowPig Hilbert invented the Hilbert polynomial :P | |
| Sep 11, 2020 at 22:34 | comment | added | Phil Tosteson | Not an expert, but the use of CM regularity is somewhat intuitive to me. The idea is that for $\mathbb P^n$ you want to parameterize graded ideals in $k[x_1, \dots, x_n]$ which a priori is a closed subset of an infinite product of Grassmannians. CM regularity lets you cut this down to the Grassmannian of monomials of degree equal to the regularity. This is just a restatement of what R. van Dobben De Bruyn said. | |
| Sep 11, 2020 at 21:26 | comment | added | Yellow Pig | Cool! What about Hilbert, did he have something to do with how Grothedieck came up with the idea of Hilbert schemes? | |
| Sep 11, 2020 at 21:24 | comment | added | R. van Dobben de Bruyn | The could be, and I would be very happy to hear if someone has something more useful to say! But at least I am somewhat convinced that I might be able to come up with them myself if I already know the EGA language well. | |
| Sep 11, 2020 at 21:22 | comment | added | Yellow Pig | Hmm, okay, thanks, maybe you are right and there indeed isn't much philosophy behind the technicalities of Hilbert schemes. | |
| Sep 11, 2020 at 21:19 | comment | added | R. van Dobben de Bruyn | As for the flattening stratification, I have never viewed this as something incredibly deep. We know that Hilbert polynomials are constant in flat families, and in general vary in a constructible way. The flattening stratification is just a neat way to package this geometrically. If you want to be philosophical about it, it's some incarnation of the (model-theoretic) idea that finite type things (like coherent sheaves) behave in a constructible way. For example, for coherent sheaves on the base itself, there is an open where it is locally free, and then proceed by Noetherian induction. | |
| Sep 11, 2020 at 21:17 | comment | added | Yellow Pig | Sure, thanks a lot, I'll take this into account. | |
| Sep 11, 2020 at 21:14 | comment | added | R. van Dobben de Bruyn | Right, I figured none of this was new information, which is part of the reason I put it in the comments. But to me it does offer most of the explanation of why CM regularity exists and is used in the construction, and even how one would come up with it. | |
| Sep 11, 2020 at 21:13 | comment | added | Yellow Pig | Thanks a lot! Well, I know that the construction of the Hilbert scheme is based on embedding into a bigger space, I guess this is what it starts with. Thanks a lot for the information though!! | |
| Sep 11, 2020 at 21:08 | comment | added | R. van Dobben de Bruyn | I have no idea if this is historical, but here is one way to look at it. If you want to represent a functor, you need to find some space you can embed everything in. For example, for moduli of smooth curves of genus $\geq 2$, you can use the tricanonical embedding to put them all in the same projective space. For subschemes of a projectivised scheme $(X,H)$, the natural thing to look at is embeddings defined by $nH$. Just like in the curve case, you need some a priori bound on what multiple of $H$ you need, and that's where Castelnuovo–Mumford regularity comes in. | |
| Sep 11, 2020 at 20:57 | history | edited | Yellow Pig | CC BY-SA 4.0 |
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| S Sep 11, 2020 at 20:51 | history | suggested | user2831784 | CC BY-SA 4.0 |
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| Sep 11, 2020 at 20:47 | review | Suggested edits | |||
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| Sep 11, 2020 at 20:33 | history | edited | Yellow Pig | CC BY-SA 4.0 |
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| Sep 11, 2020 at 20:26 | history | edited | Yellow Pig | CC BY-SA 4.0 |
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| Sep 11, 2020 at 20:08 | history | edited | Yellow Pig | CC BY-SA 4.0 |
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| Sep 11, 2020 at 20:02 | history | asked | Yellow Pig | CC BY-SA 4.0 |