# Questions tagged [hilbert-schemes]

The hilbert-schemes tag has no usage guidance.

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### Core of the Jordan quiver variety

It is known that, given the Jordan quiver, dimension vectors
$\textbf{v}=n,\textbf{w}=1$ and a stability condition $\theta<0,$ the corresponding quiver variety $\mathcal{M}_{\theta}(n,1)\cong \...

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### Hilbert schemes of points on toric surfaces

Let $\mathrm{S}$ be a smooth toric surface. The Hilbert scheme of $n$ points $\mathrm{Hilb}^n(\mathrm{S})$ inherits a torus action, but need not admit the structure of a toric variety itself. For ...

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### Is a Procesi bundle equipped with a hyperholomorphic connection?

Haiman has constructed in the paper the unusual tautological bundle $P$, called Procesi bundle, of rank $n!$ over the Hilbert schemes of points on the affine plane in the following way.
Let $H _ { n }...

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### Multiple of a flat family of subschemes is flat

Let $X$ be a fixed curve (e.g. a Noetherian, projective scheme of dimension 1, of finite type over an algebraically closed field $k$) and let $S$ be an arbitrary parameter scheme over $k$. Let $D \...

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### On non-representability of certain hom schemes

Let $k$ be an algebraically closed field. It is well-known that the isom-sheaf Isom$(\mathbb{A}^1_k,\mathbb{A}^1_k)$ is not representable by an algebraic space. (To be clear, the functor Isom$(\mathbb{...

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### Hilbert scheme of Grassmannians

Let $X=\mathbb{G}(k,n)$ be the Grassmannian of $k$-planes in $n$-space. Let $Y\subseteq X$ a subvariety and let $H$ be the connected component of the Hilbert scheme of $X$ that contains $Y$.
Is $H$ ...

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### References for Hilbert schemes over non-Archimedean valuation

Can you suggest me some suitable references to learn the theory of Hilbert polynomials (or related Hilbert schemes) in the non-Archimedean setting?
Thanks.

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

Let $X$ be a smooth and projective variety of dimension $d>1$. Let $X^{[2]}$ denote the Hilbert scheme of length two subschemes of $X$. Let $X^{(2)}:=X\times X/\mathbb{Z}_2$, where $\mathbb{Z}_2$ ...

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### Do subvarieties naturally map to the hilbert scheme of points?

Let $X$ be a smooth (complex) variety, and $V\subset X$ a reduced, normal subvariety. Fix $k\geq 0$. Then there exists an $n$ such that: for a generic point $v\in V$, we can intersect $V$ with the ...

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### Hilbert scheme of points and passing curves

It is well known that through five points on a projective plane you can pass a conic.
Q. What happens when points collide ?
More precisely: if I consider a more simple question of two points and ...

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### The scheme structure on the Hilbert scheme of an Abel-Jacobi curve

Let $C$ be a smooth curve of genus $g\geq 3$, embedded in its Jacobian $X=\textrm{Jac } C$ via an Abel map. Let $\textrm{Hilb}_1(X)$ be the Hilbert scheme of curves in $X$, and let $[C]\in\textrm{...

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### Algebraic cycles, Chow spaces and Hilbert-Chow morphisms

In the sequel, let $S$ be a scheme, and $X$ a locally of finite type algebraic space over $S$.
In his thesis ([R1-R4]), David Rydh introduces, among several others, the notion of relative cycles on $...

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### Is complete intersection a open or closed property in Hilbert schemes

Fix an integer $N$, $X$ a (smooth) complete intersection subvariety in $\mathbb{P}^N$. Denote by $P$ the Hilbert polynomial of $X$ (as a subvariety in $\mathbb{P}^N$). Consider the Hilbert scheme $\...

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### How does the “todd class operator” commute with Nakajima's q operators on Hilbert schemes of points on surfaces

Let $S$ be a surface, and $S^{[n]}$ the Hilbert scheme of $n$ points on $S$. One defines
$$
\mathbb H = \bigoplus_n H^*(S^{[n]}).
$$
One has Nakajima's operators $\mathfrak q_n(\alpha)$ (with $\alpha \...

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

What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$?
I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be ...

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### Constructing embedded families of curves with general moduli

Is there a known way to construct flat families of smooth curves $\mathcal{C}/\mathcal{B}$ which are fiberwise embedded in a family of projective varieties $\mathcal{X}/\mathcal{B}$ and which have ...

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### Irreducible but not geometrically irreducible component of Hilbert scheme

If $K$ is a field, is there an irreducible component of the Hilbert scheme ${\rm Hilb}_{\mathbb{P}^r_{K}}$ that is not geometrically irreducible?

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### An analogue of Brill-Noether for hypersurfaces?

Let $d,g,r$ be natural numbers such that $d \geq 1$, $g \geq 2$, $r \geq 3$. Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme classifying subschemes of $\mathbb{P}^r$ with Hilbert polynomial $P(x) = ...

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### A relation of convergence in Hilbert scheme to convergence in sense of currents

Let $\{X_i\}$ be a sequence of closed irreducible $k$-dimensional subvarieties of $\mathbb{C}\mathbb{P}^n$ of degree $d$ (they may be assumed to be smooth if necessary). Assume that this sequence ...

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### Equivariant Hilbert schemes of points

Let $G$ be a finite subgroup of $\mathrm{SL}(2, \mathbb{C})$ and let $X$ be the quotient surface $X = \mathrm{Spec}(\mathbb{C}[x, y]^{G})$. Denote by $\mathrm{Hilb}^r([X])$ the equivariant Hilbert ...

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### Counting Hilbert polynomials of projective varieties

EDIT. Fix $n,d,k\in\mathbb{N}$. Let us consider the set $\mathcal{P}_{n,d,k}$ of polynomials $P$ in one variable for which there exits a closed irreducible subvariety $X_P\subset \mathbb{C}\mathbb{P}^...

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### Limit of a sequence of smooth varieties in Hilbert scheme

Let $\{Z_i\}_{i=1}^\infty$ be a sequence of smooth irreducible $k$-dimensional submanifolds of $\mathbb{C}\mathbb{P}^n$ which converges to a closed subscheme $Z$ in the sense of the Hilbert scheme of $...

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### Subset of a Hilbert scheme consisting of smooth subvarieties

Let $X$ be a smooth projective variety over an algebracally closed field $k$.
(In my case $k=\mathbb{C}, X=\mathbb{P}^n$.)
Let us consider the subset of $k$-points of the Hilbert scheme $Hilb(X)$ ...

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### The non-curvilinear locus in $\textrm{Hilb}^4(\mathbb C^2)$

Let $H_n=\textrm{Hilb}^n(\mathbb C^2)$ be the Hilbert scheme of $n$ points in $\mathbb C^2$ and let $H_n^0\subset H_n$ the punctual Hilbert scheme, parametrizing subschemes entirely supported at the ...

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

It is a well known fact that a smooth cubic surface in $P^3$ contains 27 lines. One proof proceeds by moving through the parameter space $U$ of smooth cubics until one reaches an cubic that can be ...

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### A Special Case of Maximal Rank Conjecture

A special case of maximal rank conjecture states that for a general curve $C$ and general points $p_1,\dots ,p_n\in C$ the map
$$Sym^2H^0(K_C-p_1-\dots -p_n)\to H^0(K_C^{\otimes 2}-2p_1-\dots -2p_n)$$
...

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### Top intersections on the Hilbert scheme of points on a surface

The Picard group of $S^{[n]}$ is generated by the Picard group of $S$ (via a map $L \mapsto L_n$) and $E$, where $E = -\frac{B}{2}$, where $B$ is the exceptional divisor of the Hilbert Chow morphism.
...

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### If a subgroup H of a finite group G acts freely on a variety, can the G-Hilbert scheme be computed by iteration?

Let $X$ be a smooth quasi projective variety over $\mathbb{C}$. Let $G$ be a finite abelian group acting via automorphisms on $X$.
Denote by $G$-$\text{Hilb}(X)$ the subscheme of the Hilbert scheme ...

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### Hilbert scheme of a plane conic union a point

In Alex Lee's undergraduate thesis (2000), it was said that the Hilbert scheme $H_{2m+2}(\mathbb{P}^3)$ has two components $\mathcal{H}',\mathcal{H}''$, where a general point of $\mathcal{H}'$ ...

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### Transversality of quadrics containing a projective curve

Let $C$ be a curve of genus $g$ and $L$ a $g^r_d$ on it and assume that we are in the range ${r+2\choose 2}>2d-g+1$. If $C$ and $L$ are chosen to be general then by the maximal rank conjecture (...

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### Is the Chow scheme of 1-cycles the space of Cohen-Macaulay curves?

Let $C\subset X$ be a smooth irreducible curve of genus $g$, embedded in a smooth projective 3-fold $X$. So its homology class $\beta=[C]\in H_2(X)$ is an irreducible class. I want to compare two ...

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### Curves and trisecant lines

We know that rational normal curves and elliptic normal curves have no trisecant lines. For the "next" case, this is still true. That is, a nondegenerate curve of degree $d\geq 5$ and genus $2$ in $\...

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### Regarding a conjecture Fogarty proposed

In a paper by Fogarty titled "Algebraic Family On An Algebraic Surface,"
he conjectured that $\bf Hilb^n(\mathbb P^N)$ is always variety-- reduced and irreducible.
Is this still a conjecture; any ...

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### hilbert quot stacks vs schemes

What is the calculation that shows that the Hilbert or Quot functors could be represented by schemes (under various noetherian, (quasi) projectivity hypotheses), and do not require extending to the ...

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### Hilbert scheme of projectively normal elliptic curves

Consider the Hilbert scheme of degree $n$, genus $1$ curves in $\mathbb P^{n-1}$. It contains the locus of smooth curves embedded by the complete linear system of a degree $n$ divisor. Let $X_n$ be ...

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### representability of some mapping stack

Let $S$ be an Artin stack of finite type.
We assume that it contains a point as an open dense.
Is it always true that the mapping stack:
$Hom^{0}(\mathbb{P}^{1},S)$
which consists of sections ...

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

My friend, who is currently taking an algebraic geometry course from an unnamed prolific poster on MO, told me about the following bonus question on one of his problem sets a few weeks ago.
...

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

In this MO question it is stated that there is a relation between some aspects of condensed matter physics (namely the fractional quantum Hall effect) and the algebraic geometry of Hilbert schemes.
...

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### Reference Request for Hilbert Schemes

I'm a physicist working on Fractional Quantum Hall effect. The mathematical subjects of study are symmetric, translational invariant, homogeneous polynomials on $\mathbb{C}$. Very early in my study I ...

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### Hilbert schemes and moduli of ideal sheaves

Let $X$ be a smooth projective variety over $\mathbb{C}$. The Hilbert scheme on $X$ parametrizes quotients $\mathcal{O}_X \to E$ with fixed Hilbert polynomial. Let us fix the Hilbert polynomial to ...

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### Choosing a group action to do GIT of hypersurfaces

When studying GIT stability of hypersurfaces $d$ of $\mathbb P^n$ we look at the Hilbert Scheme $H=\mathbb P^N$ parametrizing homogeneous polynomials $f_d(x_0,\ldots,x_n)$ of degree $d$. There is ...

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### Standard techniques on rationally connected varieties

Is there some standard technique or approach to determine when a (irreducible) subvariety of a rationally connected variety is again rationally connected? Any reference/text dealing with this kind of ...

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### 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 ...

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### Hilbert scheme of relative subschemes of lenght 2

Let $\mathfrak X \rightarrow S$ a smooth projective family over the spectrum of a dvr. We know that $(\mathfrak X _{\eta_R})^{[2]}$ and $(\mathfrak X _{p})^{[2]}$ are smooth, where $p$ is the closed ...

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### Set of smooth curves on the Hilbert scheme is open. H

Let $H = Hilb_{d,g,r}$ be the Hilbert scheme of genus $g$ curves of degree $d$ in proyective space $\mathbb{P}^r$, over an algebraically closed field $k$.
Is it true that the set of points of $H$ ...

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### Gluing subschemes of fibers of Hilbert Scheme (in mixed characteristic)

Let $\mathfrak{X}\rightarrow Spec(R)$ be a smooth family of smooth projective varieties over a local 1-dimensional ring of mixed characteristic. Suppose that there are non-empty subschemes (locally ...

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### Göttsche's formula for non-compact complex surfaces?

Is the Göttsche's formula (Eq (2.1) of this paper) expressing the Poincare polynomial (or the Euler char version) of the Hilbert scheme of points on a projective surface valid for non-compact complex ...

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### 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 $\mathrm{Hom}_K(\mathrm{Spec}(A),\mathrm{...

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### 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 \...

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### 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 ...