The hilbert-schemes tag has no usage guidance.

**18**

votes

**1**answer

661 views

### Fuss-Catalan algebras and non-commutative Hilbert schemes

Hello, this is a question regarding Reineke's paper "Cohomology of non-commutative Hilbert schemes", http://arxiv.org/abs/math/0306185, and more precisely the formula on page 4 there (at $n=1$), ...

**17**

votes

**2**answers

1k views

### Why do flag manifolds, in the P(V_rho) embedding, look like products of P^1s?

Bert Kostant mentioned an odd fact to me some time ago. As usual (with such statements), fix a
complex, connected, reductive) Lie group $G$, with maximal torus $T$, and Weyl vector $\rho$ equal to ...

**12**

votes

**2**answers

822 views

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

In the following, let $G=\operatorname{GL}_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more ...

**10**

votes

**2**answers

443 views

### Why is Maps(X,Y) an open subfunctor of Hilb(X x Y)?

Let $X$ and $Y$ be projective schemes. Then we can define the mapping scheme between them, $\rm{Maps}(X,Y)$ as follows:
To any map $f:X\rightarrow Y$ we consider the graph $\Gamma_f$ as a closed ...

**9**

votes

**3**answers

727 views

### Reference for combinatorics of cell decomposition of the Hilbert scheme of points in the plane

It is known from either Morse theory or Bialynicki-Birula decomposition that the fixed points of a ${\mathbb{C}}^*$ action on a smooth algebraic variety over $\mathbb{C}$ determine a cell ...

**9**

votes

**2**answers

619 views

### Are spaces of holomorphic maps manifolds?

Hello,
Let $X$ and $Y$ be two smooth (probably projective) algebraic varieties defined over $\mathbf{C}$.
What is known in general about the (topological) space of holomorphic maps ...

**9**

votes

**1**answer

555 views

### There are only finitely many varieties up to deformation

Let $h$ be a polynomial. Then results of several authors (including Chow, Grothendieck, Matsusaka, Mumford, Kollar and Viehweg) imply that the moduli space of polarized varieties with Hilbert ...

**7**

votes

**3**answers

724 views

### Families of ideal sheaves: What's the correct definition?

I'm looking at Bridgeland's paper "Flops and Derived categories" and I got confused on what he meant by a family of ideal sheaves.
Let $Y$ be a scheme, and let $S$ be another scheme. A family of ...

**7**

votes

**3**answers

1k views

### Reference request: is the punctual Hilbert scheme irreducible?

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 ...

**7**

votes

**1**answer

146 views

### Smooth quadric hypersurface, Hilbert scheme is blowup of Grassmannian?

Let $Q \subset \mathbb{P}^n$ be a smooth quadric hypersurface. Where can I find a proof of/can anyone supply a proof of$$\text{Hilb}_{2m + 1}(Q) \cong \text{Bl}_{OG(3, n+1)}G(3, n+1)?$$Can we conclude ...

**7**

votes

**1**answer

329 views

### Is there a way to check if a relative Hilbert Scheme is reduced?

More specifically, suppose I have a rational curve on a complete intersection, and I know that the relative Hilbert Scheme is not smooth at the point corresponding to my rational curve. Is there any ...

**6**

votes

**4**answers

1k views

### When are Hilbert schemes smooth?

I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a ...

**6**

votes

**2**answers

814 views

### Relationship between Hilbert schemes and deformation spaces

Hi, I'm just starting to learn about deformation theory (via Hartshorne's Deformation theory, as well as Fantechi's section of FGA explained), and I feel like I'm confused about fundamental concepts. ...

**6**

votes

**2**answers

435 views

### Parameter space for complete intersections and their discriminant

Consider globally complete intersections in $\mathbb{P}^n$, of codimension $k$, of some fixed multi-degree $(d_1,\dots,d_k)$.
Is there some nice (i.e. "explicit") parameter space for them?
(even if ...

**6**

votes

**0**answers

268 views

### A question on infinitesimal deformation (related to intersection theory)

Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then ...

**5**

votes

**2**answers

192 views

### Branch locus of a 6:1 cover of the grassmannian G(1,3)

Given a general quartic surface $S$ in $\mathbf{P}^3$, there is a natural 6:1 surjective map
$\phi: Hilb^2(S) \to G(1,3)$ sending $\{P,Q\}$ to the line through them in $\mathbf{P}^3$.
Can you ...

**5**

votes

**1**answer

664 views

### Examples of nice reduced singularities on Hilbert schemes--Edited

In his "Murphy's Law" paper, Vakil showed that every "singularity type" (with a precise meaning) occurs on certain Hilbert schemes; for instance, the Hilbert scheme of nonsingular curves in projective ...

**5**

votes

**3**answers

414 views

### Irreducible “family” of relative effective divisors of a smooth morphism

Let $\pi:X\rightarrow Y$ be a smooth proper (assume projective if needed) morphism of schemes with $Y$ locally noetherian, and let $Z\subset X$ be an irreducible integral closed subscheme containing ...

**5**

votes

**1**answer

119 views

### Connectedness of sub-varieties via Hilbert polynomials

Let $X$ be a sub-variety of $\mathbb CP^n$ and let $p_X(k)$ be its Hilbert polynomial. It is well known that some basic invariants of $X$ (such as its dimension) can be read from $p_X(k)$. I am ...

**5**

votes

**1**answer

243 views

### A question on the morphism between Hilbert schemes

Let $L_1,L_2$ be two irreducible component of two different Hilbert schemes parametrizing closed subscheme in $\mathbb{P}^n$ and $\mathbb{P}^{n-1}$, respectively. Denote by $\pi_1: \mathcal{X}_1 \to ...

**5**

votes

**0**answers

136 views

### Status of Haiman's conjectures on the Isospectral Hilbert Scheme for dim X>2?

Let $X$ be a variety of arbitrary dimension, let $H$ denote the main component of the Hilbert scheme of points of $X$ (i.e, the closure of locus of reduced subschemes), and let $Z$ be the reduced ...

**5**

votes

**0**answers

465 views

### nth symmetric power of a Riemann surface and its Jacobian

Let $C$ be a Riemann surface of genus $g$, $Sym^{n}C$ be the nth symmetric power of $C$ with $n\geq 2g$, and $JC$ denote the Jacobian of $C$.
Question: Is it generally true that $Sym^{n}C\cong ...

**4**

votes

**1**answer

492 views

### Is projective morphism with projective fiber flat?

Let $X, Y$ be quasi-projective Noetherian schemes and $f:X \to Y$ be a projective surjective morphism. Assume that every fiber of $f$ is isomorphic to a projective space $\mathbb{P}^n$ for a fixed ...

**4**

votes

**1**answer

176 views

### Deformation of curves and closed immersions

Let $\pi:\mathcal{C} \to B$ be a (flat) family of complex projective schemes of pure dimension $1$ with fixed Hilbert polynomial, in particular, for some $n \ge 3$, $\mathcal{C} \hookrightarrow ...

**4**

votes

**1**answer

310 views

### Disconnectedness of Hilbert schemes of projective schemes

Let $Y$ be a projective scheme. The naive definition of a Hilbert scheme of subschemes $X$ of $Y$ would require us to projectively embed $Y$, then ask that $X$ have a fixed Hilbert polynomial $p$.
...

**4**

votes

**1**answer

331 views

### Hilbert scheme of points on a surface

Let $X$ be a complex surface and $X^{[n]}$ be the Hilbert scheme of finite analytic subspaces $Z$ for which $dimH^0(Z,\mathcal{O}_Z)=n$. I have trouble understanding $X^{[n]}$. That's what i've worked ...

**4**

votes

**1**answer

318 views

### What are the “special” strata of Sym^n(C^2)?

The affine variety $Sym^n(\mathbb{C}^2)$ has a natural quantization as a spherical rational Cherednik algebra. Thus, any primitive ideal of the rational Cherednik algebra has an corresponding ideal ...

**4**

votes

**1**answer

321 views

### Proving that the Hilbert scheme of points on $\mathbb C^2$ is smooth

On a summer school for undergraduate and graduate students Okounkov gave the following exercise (without hints): Prove that the Hilbert scheme of points on $\mathbb C^2$ is smooth.
Only a definition ...

**4**

votes

**1**answer

368 views

### Complete intersection space curves

Fix two positive integers $d, e$ and assume $d>e$. Is it true that a general degree $e$ curve which lies in a complete intersection of a degree $e$ and a smooth degree $d$ surface in $\mathbb{P}^3$ ...

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**0**answers

223 views

### “Reductive Groups and Hilbert Schemes” - Reference

Bezrukavnikov and Ginzburg have unpublished notes, 'Hilbert Schemes and Reductive Groups' (referenced here, for example): does anyone know what became of these notes? Did Bezrukavnikov-Ginzburg ...

**3**

votes

**1**answer

250 views

### Hilbert scheme of 2 points on an elliptic curve

The Hilbert scheme of 2 points on an elliptic curve $C$, $Hilb^2(C)$, has a natural structure of ruled surface, given by the map $f:Hilb^2(C) \to C$ such that $f(P,Q)=P+Q$.
What can we say about the ...

**3**

votes

**1**answer

161 views

### Upper bound on the number of generators of a local complete intersection curve in $\mathbb{P}^3$

Let $C$ be a local complete intersection curve in $\mathbb{P}^3$ (not irreducible or smooth) of degree $e$. Suppose $f_1, f_2$ (and $e_i=\deg(f_i)$) are two of the lowest degree generators of $I(C)$. ...

**3**

votes

**0**answers

277 views

### Hilbert vs Chow in nice cases

I'm trying to understand the relationship between the Hilbert schemes and Chow varieties in situations where everything is simple. Suppose that $X$ is a smooth projective variety over $\mathbb C$, ...

**3**

votes

**0**answers

159 views

### Projective schemes with a fixed hyperplane section

Let $H$ be a hyperplane in $\mathbb P^n$, and $X \subseteq H$ be a subscheme. Let $CX \subseteq \mathbb P^n$ be the cone on $X$ from a point $p \notin H$.
Let $Hilb_{CX}$ be the Hilbert scheme whose ...

**3**

votes

**0**answers

143 views

### Non-reduced flag Hilbert schemes

Let $P, Q$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Assume that $pr_2(Hilb_{Q,P})$ is positive dimensional where $pr_2$ is the natural projection map onto the second coordinate and ...

**3**

votes

**1**answer

253 views

### Upper bound on the dimension of the Hilbert scheme of space cuves

Denote by $H_{P,Q}$ the flag Hilbert scheme parametrizing a pair $(C,X)$ such that $X$ is a degree $d$ surface in $\mathbb{P}^3$ with Hilbert polynomial $Q$ and $C \subset X$ is a curve with Hilbert ...

**3**

votes

**0**answers

254 views

### Hilbert function of a Hilbert scheme

Given a Hilbert scheme $H$ of curves in $\mathbb{P}^3$ satisfying certain Hilbert polynomial, is there any way of understanding the degree or arithmetic genus of an irreducible component of the ...

**3**

votes

**0**answers

200 views

### Polarizations on $M_{0,n}$ from Kapranov's quotient constructions

In Kapranov's marvelous paper Chow quotients of Grassmannian I, he proves that $\overline{M}_{0,n}$ is isomorphic to both the Hilbert quotient and Chow quotient $(\mathbb{P}^1)^n//\text{SL}_2$. These ...

**2**

votes

**1**answer

243 views

### What is the family derived from the absolute Frobenius on the Hilbert scheme?

Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, ...

**2**

votes

**1**answer

219 views

### On the place where $\mathrm{Hilb}_{lines}^{x}(X)$ is smooth.

Let $X\subset \mathbb{P}_{\mathbb{C}}^N$ be irreducible generically smooth closed subscheme and
let $\mathrm{Hilb}_{lines}^{x}(X)$ denote
the Hilbert scheme of lines contained in $X$
and passing ...

**2**

votes

**1**answer

123 views

### Generic vs General property of reducedness in a family of projective schemes

Let $\pi:\mathcal{X} \to B$ be a flat family of projective schemes, $B$ is irreducible. Let $\mathrm{Spec} K$ be a generic point on $B$. Denote by $\mathcal{X}_K$, the pull-back of $\mathcal{X}$. This ...

**2**

votes

**1**answer

299 views

### Deformation of space curves to union of lines

Does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)? Furthermore, given a curve $C \subset \mathbb{P}^3$ ...

**2**

votes

**1**answer

204 views

### Carving out subsheaves of local hom-sheaves of stacks of categories

Recall from my previous question the definition of a local hom-sheaf of a stack of categories. I am interested in stacks of categories such that the underlying stack of groupoids is a moduli stack.
...

**2**

votes

**1**answer

216 views

### When is the Hom-scheme connected?

Suppose that $A$ and $B$ are two algebras finite over a field $K$ (which may be assumed to be separably closed, if that helps), then we know that the functor ...

**2**

votes

**1**answer

163 views

### Computing Euler Characteristics of Line Bundles on the Hilbert Scheme of n points

Let $S^{[n]}$ be the Hilbert scheme of $n$ points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...

**2**

votes

**1**answer

131 views

### A question on the Hilbert scheme $I_n(X,\beta)$

Let $X$ be a smooth projective threefold. Let $I_n(X,\beta)$ be the Hilbert scheme parametrizing subschemes $Z \subset X$ with curve class $\beta \in H_2(X,\mathbb{Z})$ and $\chi(\mathcal{O}_Z)=n$. ...

**2**

votes

**2**answers

287 views

### General degree $d$ surface in $\mathbb{P}^3$

Let $H_{d_1,g_1}, H_{d_2,g_2}$ be two Hilbert schemes of curves in $\mathbb{P}^3$ with degrees $d_1, d_2$ and genus $g_1, g_2$. Denote by $H:=H_{d_1,g_1}\times H_{d_2,g_2}$
where an element in $H$ is ...

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**0**answers

159 views

### computation with Hilbert scheme of $n$ points on $\mathbb C^2$ [closed]

How can we show that
$$\sum_{n = 0}^\infty q^n \operatorname{char}_T S^n(\mathbb C[x,y])=
\prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$
where $\operatorname{char}_T V$ denotes the character ...

**2**

votes

**0**answers

102 views

### Computing Euler Charactistics of Line bundles on Hilbert Schemes of points on Surfaces

Let $S^{[2]}$ be the Hilbert scheme of two points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...

**2**

votes

**0**answers

62 views

### calculating the Hilbert polynomial of a scheme given its primary decomposition

Given a scheme $X$ in $\mathbb{P}^n$, let $I_X$ be its, saturated, associated ideal. Suppose that a primary decomposition of this ideal is given by
$$
I_X =I_1 \cap \ldots \cap I_2
$$
I was ...