Questions tagged [hilbert-schemes]
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184
questions
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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 ...
- 231
4
votes
1
answer
289
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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 ...
- 4,966
2
votes
0
answers
142
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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 ...
- 5,607
1
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0
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129
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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
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0
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132
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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 $(\mathbb{P^1})^n$ ...
- 575
3
votes
1
answer
276
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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 ...
- 5,607
2
votes
0
answers
95
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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
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2
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345
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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
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250
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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 ...
- 1,724
2
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167
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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 ...
- 227
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1
answer
227
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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\...
- 227
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votes
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answer
128
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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 $U \subset S$,...
- 6,756
3
votes
0
answers
172
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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 ...
- 101
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0
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116
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 ...
- 1,696
2
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108
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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 ...
- 5,607
3
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179
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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 ...
- 754
2
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0
answers
216
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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
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161
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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 $n,P$ ...
- 23k
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356
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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 ...
- 463
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140
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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
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0
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73
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 ...
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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 ...
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0
answers
237
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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}(...
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0
answers
392
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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\...
- 1,471
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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 $...
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votes
1
answer
393
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 $...
- 4,966
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answers
146
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 ...
- 1,471
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135
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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 $\...
- 393
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answers
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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. ...
- 1,696
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154
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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. $\...
- 1,696
7
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answers
492
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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 ...
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answers
185
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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 ...
- 754
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174
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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 ...
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0
answers
80
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}\...
- 1,696
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142
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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 ...
- 2,691
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0
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327
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 ...
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2
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0
answers
181
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 \...
- 4,966
8
votes
1
answer
259
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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 ...
- 30.1k
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0
answers
139
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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$ ...
- 41
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217
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Proposition from Kollar's Rational Curves on Algebraic Varieties
$\DeclareMathOperator\Hom{Hom}$I have some questions on a proof of Proposition II.3.10 & notations from Rational Curves
on Algebraic Varieties by Janos Kollar (page 117).
We work in setting ...
- 4,966
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0
answers
109
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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....
- 2,379
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1
answer
503
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Tangent space to Hilbert schemes of points
Let $X$ be a smooth, projective rational surface and $Z$ be a zero-dimensional subscheme of $X$. Denote by $\mathcal{I}_Z$ the ideal sheaf of $Z$ in $X$ and $\mathcal{O}_Z$ the structure sheaf. Is it ...
- 2,096
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vote
1
answer
347
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How to understand the proof of Proposition 2.1 in the paper 'Nodes and the Hodge conjecture'?
In the Proposition 2.1 of the paper 'Nodes and the Hodge conjecture', R.P.THOMAS gives a proof to descending the Hodge conjecture into showing that every (n,n)-Hodge class in a $2n$-dimensional smooth ...
- 806
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90
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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{...
- 205
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47
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What can one say about a subscheme of a Hilbert scheme, which is covered by lines?
k= complex numbers, X/k closed subscheme of a Grassmannian, which is Plücker embedded in a projective space. a)X is simply connected b) the first Chow group (rational coefficients)of X is generated by ...
- 49
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166
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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 ...
- 171
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211
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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 ...
- 479
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0
answers
93
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Uniqueness of the scheme structure for a given Hilbert polynomial
If we have two lines in $P^3$ which are skewed, then we can take the union of those lines as a subscheme of $P^3$ in order to obtain a subscheme of $P^3$ with a Hilbert Polynomial given by $2m+2$.
...
- 11
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votes
1
answer
364
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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 \...
- 1,537
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335
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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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