Questions tagged [hilbert-schemes]

The tag has no usage guidance.

100 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
10 votes
0 answers
376 views

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 ...
ssx's user avatar
  • 2,729
9 votes
0 answers
373 views

Kähler metric on the Hilbert scheme of points on a surface

Question. Let $S$ be a non-singular complex projective surface and let $S^{[n]}$ be its Hilbert scheme of $n$ points. Is there a natural way to associate to a Kähler metric $\omega$ on $S$ a Kähler ...
Jost Schultze's user avatar
7 votes
0 answers
158 views

Is the universal object over a Hilbert scheme connected?

Hartshorne proved in his thesis that if $S$ is connected, then the Hilbert scheme $\operatorname{Hilb}^p=\operatorname{Hilb}^p(\mathbb{P}^n_S/S)$ is too (where $p\in \mathbb{Q}[z]$). Can the same be ...
Nathan Lowry's user avatar
7 votes
0 answers
536 views

Philosophical underpinnings of Grothendieck's construction of the Hilbert scheme

Long ago when I was in grad school I was told that Grothendieck's construction of the Hilbert scheme is rooted in two main technical points: Castelnuovo-Mumford regularity and Mumford flattening ...
Yellow Pig's user avatar
  • 2,490
7 votes
0 answers
179 views

Open subfunctor of Quot Functor induced by open immersion

Let $f: X \rightarrow S$ be a morphism of noetherian schemes and $\mathfrak{Q}uot_{\mathcal{E}/X/S}$ be the functor parametrizing families of quotients of $\mathcal{E}$ in the category of locally ...
lsdrs's user avatar
  • 171
7 votes
0 answers
342 views

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 ...
Xudong's user avatar
  • 143
7 votes
0 answers
434 views

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 ...
Will Sawin's user avatar
  • 135k
7 votes
0 answers
288 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 ...
dhy's user avatar
  • 5,888
6 votes
0 answers
404 views

This sum over partitions has unexpectedly nice denominators

Fix an integer $n\geq0$, a power series $\gamma \in \mathbb Q[[X]]$ with valuation 1, and a symmetric function $f$ (with coefficients in $\mathbb Q$). Now, consider the series $$ S_n = \sum_{\Lambda\...
Drew's user avatar
  • 1,469
6 votes
0 answers
160 views

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 \...
Drew's user avatar
  • 1,469
6 votes
0 answers
170 views

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 ...
asv's user avatar
  • 21.1k
6 votes
0 answers
319 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 ...
user46578's user avatar
  • 823
6 votes
0 answers
405 views

Punctual Hilbert Schemes

Let $H$ be the Hilbert scheme of Artin local rings (quotients of a power series ring $R$ in $e$ variables over $\mathbb{C}$) of length $n$. Consider the set $G\subset H$ of rings $A$ with the property ...
Mohan's user avatar
  • 6,117
5 votes
0 answers
151 views

A conjecture about sums over partitions arising from Hilbert scheme of points

$\DeclareMathOperator{\leg}{\operatorname{leg}}\DeclareMathOperator{\arm}{\operatorname{arm}}$The following situation arose from the study of some localization computations on Hilbert schemes of ...
Drew's user avatar
  • 1,469
5 votes
0 answers
155 views

The structure of the Hilbert scheme of conics contained in hypersurfaces in $\mathbb P^3$

We work over a field of characteristic $0$. Let $X\hookrightarrow\mathbb P^3$ be a geometrically integral hypersurface of degree $\delta$. It is well known that the Hilbert scheme of conics in $\...
var's user avatar
  • 403
5 votes
0 answers
151 views

Hilbert scheme of real curves

Morally speaking, my question is whether every real smooth projective curve can be deformed in as many real directions as complex directions. Let me make the question precise. Let $H$ be the Hilbert ...
Hans's user avatar
  • 2,863
5 votes
0 answers
257 views

Fibers of the Hilbert-Chow morphism vs local punctual Hilbert schemes

Let $X$ be a curve over a scheme $k$: Let $H_{n,X}$ be the punctual scheme of $X$ parametrizing finite subschemes of degree $n$, and le $\varphi_{n,X}: H_{n,X} \rightarrow X^{(n)}$ be the Hilbert-Chow ...
Raffaele C's user avatar
5 votes
0 answers
230 views

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 }...
Satoshi  Nawata's user avatar
5 votes
0 answers
244 views

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 ...
Brenin's user avatar
  • 1,514
5 votes
0 answers
726 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 JC\...
YOURS's user avatar
  • 533
4 votes
0 answers
152 views

Hilbert scheme of points concentrated in a given point

It is well known that if $X$ is a smooth surface, then the Hilbert scheme of points $X^{[n]}$ is also smooth. What about the subscheme $S_p$ of $X^{[n]}$ consisting of all schemes of finite length $Z$ ...
Learner2000's user avatar
4 votes
0 answers
123 views

Relations between double coinvariants and affine Springer fibers

Diagonal coinvariants have an interpretation from https://arxiv.org/abs/math/0201148 in terms of the Hilbert scheme. There are two recent papers https://arxiv.org/pdf/1801.09033.pdf and https://arxiv....
Yellow Pig's user avatar
  • 2,490
4 votes
0 answers
94 views

Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian

Let $\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ be the Grassmann functor, which maps a scheme $S$ to the set: $$\left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{...
BenediktK's user avatar
  • 205
4 votes
0 answers
329 views

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 $\...
Ron's user avatar
  • 2,116
4 votes
0 answers
407 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 $\mathbb C$, ...
user avatar
4 votes
0 answers
229 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 $Hilb_{Q,...
Naga Venkata's user avatar
  • 1,010
4 votes
0 answers
321 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 ...
George Melvin's user avatar
3 votes
0 answers
190 views

Virtual fundamental class of punctual Hilbert scheme of points

$\DeclareMathOperator\Hilb{Hilb}$It is well known that the Hilbert scheme $\Hilb^n(\mathbb C^3)$ has a (symmetric) perfect obstruction theory. Consider the punctual part at $0 \in \mathbb C^3$, which ...
user147163's user avatar
3 votes
0 answers
198 views

Deformations of genus g curves to 'non-reduced rational curve'

We work over the complex numbers. Fix a genus $g$. Does there exist a connected reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions? its ...
Cranium Clamp's user avatar
3 votes
0 answers
172 views

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 ...
jacob's user avatar
  • 2,814
3 votes
0 answers
121 views

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?
DCT's user avatar
  • 1,537
3 votes
0 answers
331 views

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 ...
John's user avatar
  • 31
3 votes
0 answers
126 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 ...
NotNow's user avatar
  • 103
3 votes
0 answers
226 views

When are Hilbert schemes connected by piecewise smooth curves?

Are there examples of Hilbert scheme $H$ of curves in $\mathbb{P}^3$ such that there exists an irreducible component $L$ of $H$ such that for any two points in $L$, there exist smooth projective ...
user46578's user avatar
  • 823
3 votes
0 answers
202 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 ...
Allen Knutson's user avatar
3 votes
0 answers
448 views

Deformation of a family of curves in a surface

Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for ...
Naga Venkata's user avatar
  • 1,010
3 votes
0 answers
309 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 ...
Naga Venkata's user avatar
  • 1,010
3 votes
0 answers
249 views

Smooth curve in the Hilbert flag scheme

Let $d$ be an integer greater than $0$. Let $P_2$ be the Hilbert polynomial of a degree $d$ surface in $\mathbb{P}^3$. Recall, the Hilbert flag scheme $\mathrm{Hilb}_{P_1,P_2}$ parametrizes curves $C$ ...
Naga Venkata's user avatar
  • 1,010
3 votes
0 answers
287 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 ...
Noah Giansiracusa's user avatar
2 votes
0 answers
69 views

What is happening on the second step of left mutation?

Let $X$ be a smooth Gushel-Mukai fourfold, whose semi-orthogonal decomposition is given by $$D^b(X)=\langle\mathcal{K}u(X),\mathcal{O}_X,\mathcal{U}^{\vee}_X,\mathcal{O}_X(H),\mathcal{U}^{\vee}(H)\...
user41650's user avatar
  • 1,922
2 votes
0 answers
173 views

Lifting a morphism along quotient of a group action

Let $X$ and $Y$ be complex projective varieties. Assume there is a finite group $G$ acting on $Y$ and we denote the quotient projective variety by $Y/G$. We have a morphism of $\mathcal{Hom}$-schemes ...
user127776's user avatar
  • 5,831
2 votes
0 answers
106 views

Consequences of smoothability

I have seen that there is a lot of work on studying the smoothable component of the Hilbert scheme of points $\textit{Hilb}^n(X)$ of some variety $X$. The main results are that if $\dim X \leq 2$ then ...
Aitor Iribar Lopez's user avatar
2 votes
0 answers
184 views

Cohomology of maps between Hilbert schemes

Let $S$ be a smooth complex projective surface. We consider the following two types of Hilbert schemes of $S$. The Hilbert scheme of an ample curve $D$. Suppose that $D$ is sufficiently ample, then ...
Pène Papin's user avatar
2 votes
0 answers
119 views

On non-abelian Lefschetz hyperplane theorem

This paper studies the maps of the form $Hom(X,Y)\rightarrow Hom(D,Y)$ (where $D$ is an ample divisor on $X$) and gives conditions that when it is an isomorphism. This is called non-Abelian Lefschetz ...
user127776's user avatar
  • 5,831
2 votes
0 answers
230 views

Cohomology of Beauville–Mukai varieties

The rational second cohomology of the Hilbert scheme on a K3 surface $S$ are spanned by $H^2(S,\mathbb{Q})$ plus the class of the exceptional divisor. The mapping $H^2(S, \mathbb{Q}) \to H^2(\mathrm{...
Rodion N. Déev's user avatar
2 votes
0 answers
163 views

Can components vanish without a trace?

Let $H_{P,n}$ be the Hilbert scheme of subschemes of $\mathbb{P}^n(\mathbb{C})$ with Hilbert polynomial $P\in\mathbb{Q}[t]$, and let $U_{P,n}\to H_{P,n}$ be the flat universal family. Are there $n,P$ ...
algori's user avatar
  • 23.2k
2 votes
0 answers
91 views

A Subfunctor of Quot-functor compatible with pullbacks

Let $X$ be a smooth projective irreducible algebraic curve over field $k$. For $d,r,k,m >0$ the representable Quot scheme $\mathcal {Quot}_X^{r,d}(\mathcal{O}_X(m)^k)$ is given for any test scheme $...
user267839's user avatar
  • 5,948
2 votes
0 answers
235 views

On the structure of Hilbert schemes

While studying and solving some exercises on Hilbert schemes, I've come across many problems in Hartshorne's book on deformation theory which ask the reader to show certain properties such as ...
Cranium Clamp's user avatar
2 votes
0 answers
208 views

Smoothness of Hilbert scheme of rational normal curves

I'm trying to solve Exercise 1.26 from the book "Moduli of Curves" by Harris and Morrison on page 14: Exercise (1.26) Determine the normal bundle to the rational normal curve $C \subset \...
user267839's user avatar
  • 5,948
2 votes
0 answers
165 views

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{...
Andrea Ricolfi's user avatar