22 votes

Is there a scheme parametrizing the closed subgroups of an algebraic group?

If $k$ is a field, the Hilbert functor of closed subschemes of a $k$-scheme of finite type $X$ is representable if $X$ is projective (or more generally quasiprojective if you restrict to projective ...
18 votes
Accepted

$8$-ary operation $(\mathbb{P}^2)^8 \text{ }-\to \mathbb{P}^2$, can we say anything about what this formula would look like?

This is very much a question of recent research, solved with varying degrees of generality in the papers listed below. Here is a summary. The rational map $\mathbb{P}^{2[8]}\dashrightarrow \mathbb{P}...
15 votes
Accepted

Is there a scheme parametrizing the closed subgroups of an algebraic group?

Here is an affirmative answer in the sense of algebraic spaces under a reductivity hypothesis on the subgroups, using some hard input from SGA3. (The representability by a scheme for the functor ...
  • 1,369
13 votes
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Counting Hilbert polynomials of projective varieties

Edit. I edited the answer below so that it also applies to geometrically reduced schemes of degree $d$ and pure dimension $k$. Also, the argument shows that there is a single finite set $\mathcal{P}_{...
12 votes
Accepted

If $X$ is a degree 3 smooth integral surface in $P^N$, $N > 3$, is it still true that it contains 27 lines?

If $X\subset \mathbb P^n$ is a smooth and non-degenerated variety then $$ \deg X \ge 1 + \mathrm{codim} X $$ as you can learn from Varieties of Minimal Degree by Eisenbud and Harris. Thus to ...
11 votes
Accepted

Algebraic cycles, Chow spaces and Hilbert-Chow morphisms

Mathoverflow answer In my thesis [R4], I gave an ad hoc definition of a Chow functor ($\mathrm{Chow}_r$ above) that was meaningful also in characteristic p and close to Barlet's and Angéniol's ...
  • 4,759
11 votes
Accepted

Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$

In principle, the Chow rings of Hilbert schemes of length $d$ subschemes in $\mathbb{P}^2$ are known (though it may still be a nontrivial task to extract information from the known descriptions). Here ...
10 votes
Accepted

On non-representability of certain hom schemes

Welcome, new contributor. Let $k$ be a field. Let $Y$ be a finite type, separated $k$-scheme such that the $k$-algebra $\mathcal{O}_Y(Y)$ is a $k$-vector space of infinite dimension. For instance, ...
7 votes
Accepted

Deformation of curves and closed immersions

The answer to the first question is no. In the moduli space $\mathcal{M}_{10}$ of curves of genus 10, the complete intersections $(3,3)$ in $\Bbb{P}^3$ form a strict subvariety $\mathcal{CI}$. Pick ...
  • 35.2k
7 votes
Accepted

Smooth quadric hypersurface, Hilbert scheme is blowup of Grassmannian?

I am sure there are more direct references, but it is fairly easy to prove as well. First of all, the Hilbert scheme $\text{Hilb}_{2m+1}(\mathbb{P}^n)$ is a $\mathbb{P}^5$-bundle over the ...
7 votes

Hilbert schemes and moduli of ideal sheaves

It seems to me that $$\mathscr{I}_Z^{\vee\vee}\simeq \mathscr{O}_X,$$ because the left-hand side is reflexive and isomorphic to $\mathscr{O}_X$ away from $Z$, which has codimension at least $2$ by ...
  • 21.3k
6 votes

Connectedness of Quot schemes

The simplest example when connectedness fails is the Hilbert scheme of lines on a quadric surface: $$ \mathrm{Hilb}_{1+t}(\mathbb{P}^1 \times \mathbb{P}^1) = \mathbb{P}^1 \sqcup \mathbb{P}^1, $$ where ...
  • 33.4k
5 votes
Accepted

Multiple of a flat family of subschemes is flat

I am just posting my comment as an answer. No, that is not true. Let $X$ be $\text{Spec}\ k[t,u]/\langle tu\rangle $. Let $S$ be $\text{Spec}\ k[ϵ]/\langle ϵ^2 \rangle.$ Let $I$ be $\langle t,u−ϵ\...
5 votes

Moduli spaces in applied mathematics and condensed matter physics?

Regarding question 2: FQHEs are examples of topological phases of matter, whose effective field theories are topological quantum field theories, whose quasi-particle excitations are (essentially) ...
4 votes
Accepted

Standard techniques on rationally connected varieties

Let $[x,y,z,w]$ be homogeneous coordinates on $\mathbb{P}^3$ so that $\Gamma_*(\mathcal{O}_{\mathbb{P}^3})$ equals $k[x,y,z,w]$. Let $p$ be the point $[0,0,0,1]$ in these coordinates, whose ...
4 votes

Moduli spaces in applied mathematics and condensed matter physics?

In response to [2], the classification of topological insulators (or more generally, of any gapped condensed matter, so also superconductors) relies on moduli spaces, see for example this tutorial by ...
4 votes

Reference Request for Hilbert Schemes

I am not an expert on Hilbert schemes but I would recommend the following references: Eisenbud, Harris, "the geometry of schemes". A beautiful, relatively short introduction to scheme theory. ...
  • 301
4 votes

When are Hilbert schemes smooth?

For some more examples of smooth HS see A.P.Staal: The ubiquity of smooth Hilbert schemes, arxiv AG 31.Jan. 2017.
  • 49
4 votes

Regarding a conjecture Fogarty proposed

The conjecture is located at the bottom of page 520 in: Fogarty, John Algebraic families on an algebraic surface. Amer. J. Math 90 1968 511–521. In this paper it is shown that $\mathrm{Hilb}^n(\...
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4 votes
Accepted

Tangent space to Hilbert schemes of points

See Lemma B.5.6 in A. Kuznetsov, Yu. Prokhorov, C. Shramov, "Hilbert schemes of lines and conics and automorphism groups of Fano threefolds", Japanese Journal of Mathematics, V. 13 (2018), N. 1, pp. ...
  • 33.4k
4 votes
Accepted

tangent bundle of Hilbert schemes of points on a projective surface

The tangent bundle on $S^{[n]}$ is not quite isomorphic to $(T_S)^{[n]}$, rather by Theorem B of Stapleton's paper listed below there is an injection of the former into the latter. Stapleton, David, ...
  • 1,456
3 votes

Curves and trisecant lines

Edit. Actually, every nondegenerate curve in $\mathbb{P}^3$ except for twisted cubics and elliptic quartics has a $1$-parameter family of trisecant lines: Curve of 3-secant lines I am just expanding ...
3 votes

Moduli spaces in applied mathematics and condensed matter physics?

You can consider Nekrasov-Shatashvili for application to Toda chain or Calagero-Moser type systems. That nominally looks like a string or high energy particle link, but those systems are integrable ...
3 votes
Accepted

When is the Hom-scheme connected?

I only just noticed this question. I agree with Andrew Stout, but I am afraid that I disagree with Dmitry Vaintrob. To make Dmitry's example precise, assume $B$ is a graded, Artinian (commutative, ...
3 votes

Moduli spaces in applied mathematics and condensed matter physics?

Re: q. 3), consider the geometry of the space of phylogenetic trees.
3 votes
Accepted

Degrees of syzygies of points in $\mathbb P^2$

I asked David Eisenbud and was told about Exercises 12 and 13 of Chapter 3 in his book "Geometry of Syzygies". Putting together, they show that for a generic set of $n$ points $X$ in $\mathbb P^2$, ...
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3 votes
Accepted

irreducibility punctual Hilbert scheme of relative subschemes of length $2$

If I understand the question correctly, the subset $\mathcal{H}_U$ is the closure of the locus parametrizing two distinct points. Over this locus, the Hilbert-Chow morphism $\pi$ is an isomorphism so ...
  • 2,850
3 votes
Accepted

When Hom scheme has projective components?

Let $k$ be an algebraically closed field of characteristic zero and let $X$ be a projective variety over $k$. To try and answer your first two questions, let us try formalize the property that Hom-...
2 votes

Göttsche's formula for non-compact complex surfaces?

Yes. This is Theorem 5.2.1 of The Douady space of a complex surface by de Cataldo and Migliorini (Math.Ann.)
  • 2,899
2 votes

Families of curves with "almost-general" moduli

I realize this is an old question, but I only now noticed it. I do not know why you say that the Brill-Noether theorem implies that there is a unique component of the Hilbert scheme that dominates ...

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