Timeline for Ideals whose alebraic variety is a singleton
Current License: CC BY-SA 4.0
13 events
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| Apr 3, 2023 at 15:10 | comment | added | Pii_jhi | Thank you for yours answers it helps a lot ! I checked the literature and its really wonderful. From what I gathered I'm interested in fact in the "smoothable" component of the Hilbert scheme (i.e. specializations of $p$ distincts points) and it has an explicit construction as a blow up of the diagonal of $(\mathbb{R}^q)^p$ quotiented by the symmetric group action. I will keep reading papers and if I'm struggling or that I need some intuition I will ask the question again in better terms. | |
| Apr 3, 2023 at 13:42 | comment | added | Gro-Tsen | PS: Don't let the term “subscheme” frighten you: a subscheme of $\mathbb{A}^n_{\mathbb{C}}$ is just given by an ideal of $\mathbb{C}[t_1,\ldots,t_n]$, it is zero-dimensional iff the quotient is a finite-dimensional vector space, in which case the “length” refers to the dimension of the quotient. (But you probably inferred all this already.) | |
| Apr 3, 2023 at 13:39 | comment | added | Gro-Tsen | [contd.] I suggest you try reading the beginning of Göttsche's book I mentioned in an earlier comment (especially §2.1) and Iarrobino's paper I just cited, and perhaps search for other papers citing them, and if they don't contain what you want, re-ask the question on MathOverflow with terms like “Hilbert scheme of $0$-dimensional subschemes” and the notation from the references in paper: this will likely draw more attention from people more knowledgeable than I am. | |
| Apr 3, 2023 at 13:33 | comment | added | Gro-Tsen | “Not being specialization of $p$ distinct points” means what you guessed: a reference for this is Iarrobino, “Reducibility of the families…” Invent. Math 15 (1972) 72–77. It's essentially a dimension-count argument: the Hilbert space in question has a larger dimension than the space of $p$ distinct points, so it must contain other components. That even the dimension count isn't easy suggest that its exact value isn't known, but I'm not sure. Maybe try re-asking the question with the “right” terminology. [contd.] | |
| Apr 3, 2023 at 9:01 | comment | added | Pii_jhi | I still don't need much about these spaces, only their dimension and maybe a smooth/stratified manifold structure would already be nice. My problem is very simple, I have a function that depends on p points, whose expression is singular when points collapses and I try to understand the singularity. There is I believe others path to solve my problem than by solving explicitly the singularity, but knowing that its hard is a very interesting property :) | |
| Apr 3, 2023 at 8:55 | comment | added | Pii_jhi | I don,t know anything about schemes but this indeed seems exactly to be the object that I need. What is the intuition behind "not beeing specializations of $p$ distincts points" ? Does it mean that one cannot find a sequence of distinct point such that the associated "ideal" converge to the limit ideal we are considering ? I will try to find good references on this topic that explains things without too much algebraic geometry... From what I gather the case q=1,2 are well understood and are close to intuition, but not higher dimensional spaces. | |
| Mar 31, 2023 at 22:30 | comment | added | Gro-Tsen | (In fact, if I remember correctly, of the the surprises about these objects is that, for $p$ large enough, there exist entire components of this Hilbert scheme which are not specializations of $p$ distinct points. I think this result is by Iarrobino, but I don't remember where I learned it.) | |
| Mar 31, 2023 at 22:27 | comment | added | Gro-Tsen | In this case, a keyword which you should try searching for if you do not already know about it is “Hilbert scheme of zero-dimensional subschemes” (there is a book by Lothar Göttsche with this title) or “punctual Hilbert schemes” (there is a book by Anthony Iarrobino with this title). What you call $E_p$ is the “Hilbert scheme of $0$-dimensional subschemes of length $p$ supported at the origin” (or some analogous description). And unfortunately, it tends to be not at all smooth, nor even irreducible. | |
| Mar 31, 2023 at 15:43 | comment | added | Pii_jhi | Thank you for your answer, indeed it makes sense to replace $\mathbb{R}$ by $\mathbb{C}$, i edited my question. My question is indeed really geometrical in nature. I want to understand "how" $p$ points can be 'infinitely' close to each other. If a generic sequence of polynomial cancels on $p$ distinct points that collapse to a single point, what can I say about the limiting polynomial if it exists ? Well it has to lie in some $E_p$, so now the goal is to understand who is $E_p$. I figure it's been done already but I cannot find any reference. | |
| Mar 31, 2023 at 15:34 | history | edited | Pii_jhi | CC BY-SA 4.0 |
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| Mar 31, 2023 at 13:29 | comment | added | Gro-Tsen | (contd.) So: either your questions are really geometrical in nature, in which case you should fix them (change $\mathbb{R}$ to $\mathbb{C}$ in the definition of $V(I)$ and maybe elsewhere as well; so then $E$ is the set of ideals having $(X_1,\ldots,X_q)$ as radical); or alternatively, if you really mean $\mathbb{R}$ then you should explain $\operatorname{mult}(I)$ better, and start by stating what you already know about ideals such that $V(I)$ has no real point. | |
| Mar 31, 2023 at 13:25 | comment | added | Gro-Tsen | Are you sure you wish to ask about real ideals $I$ whose only real point is the origin rather than those whose only complex (=geometric) point is the origin? For example, with the definition you wrote, the ideal $(X_1^2+X_2^2)$ in $\mathbb{R}[X_1,X_2]$ is in $E$, although it is not zero-dimensional in any way, and in particular $\operatorname{mult}(I)$ is infinite, which seems to go against the fact that you seem to want to know geometric things about zero-dimensional ideals. So I think your question needs fixing. | |
| Mar 31, 2023 at 12:47 | history | asked | Pii_jhi | CC BY-SA 4.0 |