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Long ago when I was in grad school I was told that Grothendieck's construction of the Hilbert scheme is rooted in two main technical points: Castelnuovo-Mumford regularity and Mumford flattening stratification. In grad school I made sure to learn these two constructions very carefully, and I have been dealing with the Hilbert scheme (mostly of points in the plane) closely in my work since then, but I still don't understand at all on a philosophical level why the Hilbert scheme exists and why these two technical constructions play the main role in its existence. Could someone enlighten me regarding these questions? In general, why is the subject of Hilbert schemes so insanely technical if they are so incredibly useful in mathematics, and what could be a 'correct' point of view on Hilbert schemes which would make them easier to understand from the philosophical perspective? What would be good suggestions for literature which could help to understand the technicalities of the subject of Hilbert schemes on a more philosophical level?

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    $\begingroup$ 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. $\endgroup$ Sep 11 '20 at 21:08
  • $\begingroup$ 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!! $\endgroup$
    – Yellow Pig
    Sep 11 '20 at 21:13
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    $\begingroup$ 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. $\endgroup$ Sep 11 '20 at 22:34
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    $\begingroup$ 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. $\endgroup$
    – Balazs
    Sep 12 '20 at 10:30
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    $\begingroup$ 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. $\endgroup$ Sep 13 '20 at 1:11

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