19
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}...

14
votes

Accepted

### 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}_{...

Community wiki

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 ...

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, ...

Community wiki

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 ...

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\...

Community wiki

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) ...

Community wiki

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 ...

Community wiki

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. ...

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(\...

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.

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. ...

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, ...

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 ...

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 $\mathbb P^2$, ...

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 ...

Community wiki

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 ...

Community wiki

3
votes

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

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

Community wiki

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

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

If we work with algebraic varieties instead of schemes, then there is a positive answer: I described an ind-variety parametrising families of connected algebraic subgroups of an algebraic group $G$, ...

2
votes

### Hilbert scheme of points and passing curves

If you take any zero-dimensional subscheme Z of P^2 of length 5, the only way two conics can both contain it is if they have a common component, which must then be a line L, hence they must be of the ...

2
votes

Accepted

### Construction of an atlas for the moduli stack $\mathcal{Bun}_X^{n,d}$ in F. Neumann's 'Algebraic Stacks and Moduli of Vector Bundles'

I don't know what is going on exactly (misprints?), but here are some ideas:
If you take a point of $q\in R_m$ (i.e. $U=Spec(k)$) defined by a sequence
$0 \rightarrow G \rightarrow \mathcal{O}_X^{P(m)}...

2
votes

Accepted

### When is the morphism from the Hilbert scheme to the moduli scheme of stable sheaves an isomorphism?

This works for any smooth projective variety $X$ under the assumption
$$
\mathrm{Pic}^0(X) = 0
$$
and any $Z$ of codimension at least 2. For the proof see Lemma B.5.6 in Kuznetsov, Alexander G.; ...

1
vote

### The weight of a weighted filtration is given (for large $m$) by a polynomial

I'm aware this is an old question, but I'm answering it for the benefit of anyone who comes across this question in the future.
There are not one but two proofs of this result in the paper Uniform $K$-...

1
vote

### How to understand the proof of Proposition 2.1 in the paper 'Nodes and the Hodge conjecture'?

As R.P.Thomas told me, the relative Hilbert scheme there is the usual one, and the section 4 of the paper A remark on singularities of primitive cohomology classes gives a more explicit refinement.

1
vote

### Core of the Jordan quiver variety

Actually, the answers to 1 and 3 are mentioned in Nakajima's book on Hilbert Schemes, in Exercise 5.15, and later proved in the paper https://arxiv.org/pdf/math/0311058.pdf by Nakajima and Yoshioka. ...

1
vote

### Exceptional divisor of the Hilbert-Chow morphism of the punctual Hilbert scheme

Note that $\mathrm{E}$ is the branch divisor of the covering $p:\mathrm{Z}\rightarrow\mathrm{X}^{[n]}$, where $\mathrm{Z}\subset\mathrm{X}\times\mathrm{X}^{[n]}$ is the universal subscheme. Hence $-...

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