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The punctual Hilbert scheme in dimension $d$ parameterizes ideals $I$ of codimension $n$ in $k[x_1,\dots, x_d]$ which are contained in some power of the ideal $(x_1,\dots, x_d)$. In other words, it is the Hilbert scheme of $n$ points supported at the origin in $\mathbb A^d$.

Can anybody give me a reference for this object? In particular, I'd like to know if it is irreducible for arbitrary $d$.


A related question: the curvilinear Hilbert scheme parameterizes ideals $I$ of codimension $n$ in $k[x_1,\dots, x_d]$ such that $I\not\subset(x_1,\dots, x_d)^{n-1}$. This object is irreducible; in fact, it's covered by dense open subschemes isomorphic to $\mathbb A^m$ for some $m$. Does anybody know of a reference that calculates what these spaces are? For $n=2$, it's $\mathbb P^{d-1}$. For $n=3$, I think it's the total space of $\mathcal O_{\mathbb P^{d-1}}(1)$.

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While I prefer your use of the terminology, note well that some people take "punctual Hilbert scheme" just to mean that it is of points, rather than that they are at the origin. –  Allen Knutson Apr 4 '10 at 12:23
    
I actually worried that might be the case as I was writing this question, but hopefully I was clear enough about what I meant that I didn't cause too much confusion. –  Anton Geraschenko Apr 4 '10 at 17:18
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3 Answers

up vote 8 down vote accepted

It is certainly not irreducible if n=8 and d>3. This is analyzed nicely in the paper

Hilbert schemes of 8 points, Dustin A. Cartwright, Daniel Erman, Mauricio Velasco, Bianca Viray available at http://arxiv.org/abs/0803.0341 (and I think published in ANT).

From there you can look at the references, especially I think the first paper studying in detail this problem was

Anthony Iarrobino. Reducibility of the family of 0-dimensional schemes on a variety. Inventiones Math., 15:72–77, 1972.

Note that, at least the arXiv reference above, deals with subschemes of $\mathbb{A}^d$ not necessarily supported at the origin. On the other hand, since for $n \leq 8$ the above is the only non-irreducible example, the "new" component must be supported at the origin. Indeed in the case n=8 and d=4, the extra component is a product of a Grassmannian and an affine space.

EDIT: Let me expand on my initial answer. First a general remark: if a point on a scheme is non-singular, then it lies on a unique irreducible component of that scheme. Now, in the usual (not the punctual) Hilbert scheme of 8 points in A^4 there are two components, one being the closure of the component consisting of 8 distinct points, and one supported entirely at a single point (the point being allowed to vary; by translating the support, we can assume the point is the origin). In the arXiv paper they show that there are points supported at a single point that are non-singular and smoothable in the full Hilbert scheme (Section 4.4). It follows that such points cannot lie in the component whose points consist entirely of subschemes supported at the origin. We conclude that the punctual Hilbert scheme contains the "non-smoothable" component (necessarily supported at the origin, this one being a minimal example of a reducible Hilbert scheme), as well as more points, corresponding to the points described above (the ones appearing in Section 4.4). In particular, the punctual Hilbert scheme is not irreducible. Hope this clears up the doubts (or that it exposes my mistake!).

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Is the first paper about punctual Hilbert schemes? –  Hailong Dao Apr 4 '10 at 7:45
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this answers the wrong question. –  Dan Petersen Apr 4 '10 at 7:50
    
Maybe I am confused, but it seems to me that Theorem 1.3 addresses the intersection of the two components, one of which is necessarily supported at the origin, since it is a "minimal" example of irreducible Hilbert scheme. Since the intersection is a divisor in the smaller component, it should follow that there are two components supported at the origin. Either this, or I have not understood the question/paper! –  damiano Apr 4 '10 at 7:57
    
It looks like that paper is about the whole Hilbert scheme, but my question is about the Hilbert scheme of points supported at the origin, which is a closed subscheme of the whole Hilbert scheme. My understanding of your comment is that the paper shows that there are multiple components of the Hilbert scheme which meet the punctual Hilbert scheme, but I don't follow the argument for why the punctual Hilbert scheme is itself reducible. –  Anton Geraschenko Apr 4 '10 at 8:07
    
You would probably be satisfied by finding a point in the punctual Hilbert scheme that is a limit of distinct points in the full Hilbert scheme and that is smooth in that component, right? Such examples seem to appear in Section 4.4 in the arXiv paper. –  damiano Apr 4 '10 at 8:44
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Dear Anton, my ex-colleague Michel Granger wrote his thesis on the punctual Hilbert scheme you are asking about. It is rather technical and I haven't read it. It has been published since as a "Mémoire de la S.M.F."( the French equivalent of the A.M.S.) I hope you can find something of interest for you in it .

http://archive.numdam.org/ARCHIVE/MSMF/MSMF_1983_2_8_/MSMF_1983_2_8__1_0/MSMF_1983_2_8__1_0.pdf

[It is in French but as has often be stated, not by me but by by anglophone MathOverflowers, mathematical French should be no problem]

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Thanks Georges. Looking over it now, it doesn't appear to say anything about reducibility of the general punctual Hilbert scheme, but I'll definitely look over it more carefully. This caught my eye: the last section shows irreducibility of the Hilbert scheme of fat points in the plane contained in the p-th infinitesimal neighborhood of the reduced point. –  Anton Geraschenko Apr 4 '10 at 17:11
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(I cannot post comments, regard this as one.)

It was correctly suggested in the comments to damiano's answer that the case of dimension d=2 was settled in the 70s, here are references:

  • I believe the first proof is that of Briancon [Description de $Hilb^n(C\{x,y\})$, Invent. Math. 41 (1977)]
  • Ellingsrud and Strømme [On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987)] have an argument involving a cell decomposition, where there is only one cell of the expected dimension (by a separate argument, any extra component cannot have smaller dimension)
  • Perhaps the state of the art is the inductive argument by Ellingsrud and Lehn [Irreducibility of the punctual quotient scheme of a surface, Ark. Mat. 37 (1999)], building on previous work by Ellingsrud and Strømme.
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Probably the OP was interested (only?) in char. 0, which is the case treated by Briancon. Tony Iarrobino [Punctual Hilbert schemes. Mem. Amer. Math. Soc. 10 (1977)] proved the irreducibility in char p>0 for d = 2 and p>n. More recently, Sasha Premet proved this irreducibilty (for d=2) for all alg. closed fields [Nilpotent commuting varieties of reductive Lie algebras. Invent. Math. 154 (2003)]; his argument uses an observation of of Baranovskiĭ (relating the "nilpotent commuting variety" with the punctual Hilbert scheme). Premet's approach seems to be a simplification already in char. 0. –  George McNinch Jun 6 '10 at 18:02
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