# Tag Info

### 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 ...
Accepted

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 ...
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
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 ...
• 16.7k
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, ...
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
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 ...

### 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

### 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
Accepted

• 8,134
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
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

### 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 ...

### 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 ...
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, ...

### Moduli spaces in applied mathematics and condensed matter physics?

Re: q. 3), consider the geometry of the space of phylogenetic trees.
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 $\mathbb P^2$, ...
• 29.9k
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
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-...
• 9,087

### 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

### 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 ...

Only top scored, non community-wiki answers of a minimum length are eligible