Questions tagged [hilbert-schemes]

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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 $\mathbb{P}^2$...
gigi's user avatar
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1 vote
1 answer
207 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 ...
Stephen McKean's user avatar
1 vote
0 answers
139 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 ...
user267839's user avatar
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2 votes
1 answer
158 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 $\mathcal{O}_X$. ...
Display Name'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
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7 votes
0 answers
159 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
4 votes
1 answer
323 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 ...
user267839's user avatar
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2 votes
0 answers
176 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
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1 vote
0 answers
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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 ...
BAI's user avatar
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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 $(\mathbb{P^1})^n$ ...
tota's user avatar
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3 votes
1 answer
356 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 ...
user127776's user avatar
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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
5 votes
2 answers
415 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 $P$. ...
PMCosmin's user avatar
1 vote
0 answers
290 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 ...
Lao-tzu's user avatar
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2 votes
0 answers
186 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
1 answer
321 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\...
Pène Papin's user avatar
4 votes
1 answer
146 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 $U \subset S$,...
Libli's user avatar
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3 votes
0 answers
191 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
1 vote
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140 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 ...
user41650's user avatar
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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
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3 votes
0 answers
199 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
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
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9 votes
0 answers
375 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
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0 answers
144 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 :)!
Roxana's user avatar
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2 votes
1 answer
117 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 ...
merlino's user avatar
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5 votes
1 answer
450 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 ...
Malkoun's user avatar
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1 vote
0 answers
242 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}(...
Roxana's user avatar
  • 519
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
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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
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5 votes
1 answer
457 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 $...
user267839's user avatar
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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
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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
1 vote
0 answers
84 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. ...
user41650's user avatar
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1 vote
0 answers
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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. $\...
user41650's user avatar
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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
2 votes
0 answers
239 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
0 votes
0 answers
199 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 ...
user267839's user avatar
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1 vote
0 answers
84 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}\...
user41650's user avatar
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5 votes
0 answers
152 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
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0 votes
0 answers
369 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 ...
user267839's user avatar
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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
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8 votes
1 answer
274 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 ...
Hailong Dao's user avatar
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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
1 vote
0 answers
229 views

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 ...
user267839's user avatar
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4 votes
0 answers
124 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
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4 votes
1 answer
536 views

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 ...
Ron's user avatar
  • 2,126
1 vote
1 answer
361 views

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 ...
Bonbon's user avatar
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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
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1 vote
0 answers
51 views

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 ...
G.G.'s user avatar
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