Questions tagged [hilbert-schemes]
The hilbert-schemes tag has no usage guidance.
196 questions
1
vote
0
answers
108
views
L.c.i locus of Hilbert scheme of points on singular varieties
Let $X$ be an algebraic variety over $\mathbb{C}$. What can we say about the l.c.i. locus of $\text{Hilb}^n(X)$?
When $X$ is smooth, it is well-known that the l.c.i. locus of $\text{Hilb}^n(X)$ is ...
- 385
1
vote
0
answers
84
views
Relation between quot scheme of birational curve
I am very new to algebraic geometry. Currently reading about Hilbert and quot scheme. I want to know more about the structure and properties of Hilbert and quot schemes over curves. My question is the ...
- 613
1
vote
0
answers
102
views
weak (?) valuative criterion for properness
In the article "On the Kodaira Dimension of the Moduli Space of Curves" by J. Harris and D. Mumford, to prove that
$\overline{H}_{k,b}$ is proper over Spec $\mathbb{C}$, the authors refer to ...
- 560
0
votes
0
answers
158
views
Understanding the Hilbert scheme of subvarieties of $\mathbb{CP}^n$
EDIT: migrated to MSE.
I am looking to get a more concrete understanding of the Hilbert scheme of projective subvarieties, specifically over $\mathbb{C}$, and to obtain good references on this subject....
- 1,763
3
votes
0
answers
144
views
Curves on the Hilbert scheme of points on surfaces
Suppose $X$ is a smooth projective surface over $\mathbb{C}$ with irregularity $0$ $(q_1(X)=0)$. I want to understand the curves on the Hilbert scheme of $n$-points on $X$.
By the work of Fogarty, we ...
- 335
1
vote
0
answers
128
views
Classify all open affine subschemes of a projective variety
Currently I am pondering a question from algebraic geometry that can be stated in very simple terms: Let $X \subseteq P^n_k$ be a projective variety, subscheme of projective $n$-space over an ...
- 708
4
votes
1
answer
173
views
Nef cone of Hilbert scheme of $n$ points
Suppose $\operatorname{Nef}(X)$ is a rational polyhedron with extremal rays $\{F_i\}_i$. Now, consider the Hilbert scheme of $n$ points $X^{[n]}$ and the embedding $\operatorname{Nef}(X)\subset \...
- 335
1
vote
0
answers
52
views
Symmetric 0-dimensional schemes with generic Hilbert function and Grassmannians
I've came across this problem while thinking about some properties of fat schemes.
Let me give you an explicit (motivating) example:
We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
- 1,343
1
vote
1
answer
224
views
Hilbert scheme of points on an arithmetic surface
$\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Spec{Spec}$Let $X$ be a smooth surface over a field $k$. Fogarty proved that the Hilbert scheme of points $\Hilb^n(X)$ is regular. My primary ...
- 573
1
vote
0
answers
169
views
Étaleness of Isom scheme $\operatorname{Isom}_S(X,Y)$
Let $S$ be a quasi-projective scheme over base field $k$ and $X, Y$ two finite étale schemes over $S$ and assume we are in situation we know that the isom space $\operatorname{Isom}_S(X,Y)$ exists as ...
- 6,018
2
votes
1
answer
182
views
When is the morphism from the Hilbert scheme to the moduli scheme of stable sheaves an isomorphism?
Consider over $\mathbb{C}$. Let $(X,\mathcal{O}(1))$ be a smooth projective scheme with an ample polarisation. Let $P(t):=\chi(X,\mathcal{O}(t))$ denote the Hilbert polynomial of $\mathcal{O}_X$. ...
- 930
2
votes
0
answers
75
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)\...
- 1,982
7
votes
0
answers
178
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 ...
- 333
4
votes
1
answer
359
views
Construct morphisms of schemes on level of associated functors
I have a general question about techniques used in @Emerton's proof, sketched below, in the answer to $\mathbb{P}^n$ is simply connected.
Given a finite étale map $\pi: Y \to \mathbb P^n$ (we regard ...
- 6,018
2
votes
0
answers
210
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 ...
- 5,901
1
vote
0
answers
185
views
Cycle class/cohomology class of subvarieties in flat families
Let $X$ be a projective variety over $\mathbb C$ and $T$ an irreducible projective $\mathbb C$-scheme.
Let $a,b$ be closed points of $T$.
Suppose we have a flat family $Z\to X\times T\to T$ such that ...
- 111
1
vote
0
answers
146
views
Question regarding Hilbert scheme of points
$\DeclareMathOperator\SL{SL}$Let us consider $\SL(2,\mathbb{C})$ quotients of $(\mathbb{P^1})^n$ in the following sense. We consider diagonal action of $\SL(2,\mathbb{C})$ over $(\mathbb{P^1})^n$ ...
- 585
3
votes
1
answer
412
views
When Hom scheme has projective components?
The Hom scheme of two projective varieties over some field is constructed as an open subfunctor of the Hilbert scheme of the product of the two schemes by Grothendieck. So it is a countable union of ...
- 5,901
2
votes
0
answers
123
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 ...
5
votes
2
answers
461
views
Connectedness of Quot schemes
Let $X$ be a connected projective scheme over $\mathbb{C}$ and $E$ a coherent sheaf on $X$. Consider the Quot scheme $\operatorname{Quot}_X(E,P)$ of quotients of $E$ of fixed Hilbert polynomial $P$. ...
- 51
1
vote
0
answers
324
views
On construction of Hilbert and Quot schemes
I have some questions regarding the strategy of the proof of the existence of Hilbert and Quot schemes (I will focus on the latter since it's more general), as in the book Fundamental Algebraic ...
- 1,906
2
votes
0
answers
196
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 ...
- 279
2
votes
1
answer
377
views
tangent bundle of Hilbert schemes of points on a projective surface
Let $S$ be a smooth projective surface. We denote $S^{[n]}$ the Hilbert scheme of artinian subschemes in $S$ of length $n$, which is a smooth projective variety of dimension $2n$ by Fogarty. Let $I\...
- 279
4
votes
1
answer
154
views
irreducibility punctual Hilbert scheme of relative subschemes of length $2$
Let $X$ be an irreducible projective variety over $\mathbb{C}$ (note that I do not assume $X$ smooth) and let $ p : X \longrightarrow S$ be a projective surjective morphism. For any open $U \subset S$,...
- 7,300
3
votes
0
answers
201
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 ...
- 111
1
vote
0
answers
148
views
Dimension of Hilbert scheme of curves on Gushel-Mukai varieties
I have several questions on Hilbert scheme of Gushel-Mukai varieties. Let $X$ be a Gushel-Mukai fourfold and let $\mathcal{H}_3$ be Hilbert scheme of twisted cubics. I was wondering what is the ...
- 1,982
2
votes
0
answers
125
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 ...
- 5,901
3
votes
0
answers
222
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 ...
- 936
2
votes
0
answers
241
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{...
- 427
2
votes
0
answers
167
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$ ...
- 23.5k
9
votes
0
answers
387
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 ...
- 473
0
votes
0
answers
145
views
Chow countability argument
I would like to know what the "Chow countability argument or HIlbert schemes countability argument" says in order to finish an exercise. Any reference will also be very useful :)!
- 519
2
votes
1
answer
118
views
The weight of a weighted filtration is given (for large $m$) by a polynomial
Let $I$ be an homogeneous ideal of $k[x_0, \dots, x_n]$. Suppose to give integral weights $\lambda_0, \dots, \lambda_n$ to $x_0, \dots, x_n$. We assign a weight to every homogeneous polynomial of ...
- 45
5
votes
1
answer
486
views
Which completion of the configuration space of $n$ distinct points in $\mathbb{R}^d$ is better suited for numerical analysis?
(My original post starts here, and ends right before the Edit part. I am keeping it so that the comments and answer make sense, but what I am really interested in is what is in the Edit section.)
My ...
- 5,215
1
vote
0
answers
254
views
Construction of the Hilbert Scheme
I am reading the book "Rational Curves on Algebraic Varieties" of János Kollár. Definition-Proposition 1.2, begin like this:
Let $g:Y\rightarrow Z$ be a projective morphism and $\mathcal{O}(...
- 519
6
votes
0
answers
405
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\...
- 1,509
2
votes
0
answers
92
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 $...
- 6,018
5
votes
1
answer
504
views
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'm reading Frank Neumann's "Algebraic Stacks and Moduli of
Vector Bundles" and have some problems to understand
a construction from the proof of:
Theorem 2.67. (page 81) The moduli stack $...
- 6,018
6
votes
0
answers
152
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 ...
- 1,509
5
votes
0
answers
162
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 $\...
- 403
1
vote
0
answers
88
views
Is there a direct way to show Fano surface of lines and conics on the pairs of Fano threefolds isomorphic?
I am considering the following setting:
Let $(Y_d, X_{4d+2})$ be the pair of degree $d$ and index 2 Fano threefold $Y_d$ and degree $4d+2$ index 1 Fano threefold and both of them are Picard number 1. ...
- 1,982
1
vote
0
answers
171
views
Fano surface of conics on Gushel-Mukai threefolds
Let $X$ be a smooth Gushel-Mukai threefold, there are following four cases:
$X_1$ is a special Gushel-Mukai with branch locus $\mathcal{B}$ on $Y_5$ general, i.e, it does contain any line or conic. $\...
- 1,982
7
votes
0
answers
566
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 ...
- 2,964
2
votes
0
answers
275
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 ...
- 936
0
votes
0
answers
220
views
Use of flattening stratification (from Nitsure's construction of Hilbert and Quot schemes)
I study Nitin Nitsures paper on the Construction of Hilbert and Quot Schemes and not understand the propetry (F) completely:
In previous chapter (Embedding Quot into Grassmanian) it was
proved that ...
- 6,018
1
vote
0
answers
88
views
How to show a contraction of singular moduli space is projective?
Let $\mathcal{H}$ be a certain kind of Hilbert scheme of curves on some smooth projective variety $X$ and $\mathcal{H}$ is projective and irreducible of dimension $3$. There is a divisor $\mathcal{D}\...
- 1,982
5
votes
0
answers
161
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 ...
- 3,031
0
votes
0
answers
403
views
hypersurface of degree d Hilbert polynomial
I am in trouble to solve part (2) Exercise 1.13 from "Moduli of Curves"
by Harris and Morrison on page 9:
Exercise (1.13)
2) Fix a subscheme $X \subset \mathbb{P}^r$. Show, by taking ...
- 6,018
2
votes
0
answers
248
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 \...
- 6,018
8
votes
1
answer
285
views
Degrees of syzygies of points in $\mathbb P^2$
Let $X$ be a collection of points in $\mathbb P^2$ over the complex numbers. Let $I_X$ be the defining ideal. I am interested in knowing when:
The syzygies of $I_X$ contains no linear forms. Since ...
- 30.5k