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# Questions tagged [hilbert-schemes]

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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 ...
289 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 ...
142 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 ...
1 vote
129 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 ...
1 vote
132 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 $(\mathbb{P^1})^n$ ...
276 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 ...
95 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 ...
345 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 $P$. ...
1 vote
250 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 ...
167 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 ...
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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 $n,P$ ...
356 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 ...
140 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 :)!
1 vote
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### 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 ...
405 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 ...
1 vote
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 $\mathcal{O}(... 6 votes 0 answers 392 views ### This sum over partitions has unexpectedly nice denominators Fix an integer$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\... 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 ... 5 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 ... 5 votes 0 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 ... 5 votes 0 answers 135 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 \... 1 vote 0 answers 75 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 vote 0 answers 154 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. \... 7 votes 0 answers 492 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 votes 0 answers 185 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 ... 0 votes 0 answers 174 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 ... 1 vote 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}\... 5 votes 0 answers 142 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 ... 0 votes 0 answers 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 ... 2 votes 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 \... 8 votes 1 answer 259 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 ... 4 votes 0 answers 139 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 ... 1 vote 0 answers 217 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 ... 4 votes 0 answers 109 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.... 4 votes 1 answer 503 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 ... 1 vote 1 answer 347 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 ... 4 votes 0 answers 90 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{... 1 vote 0 answers 47 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 ... 7 votes 0 answers 166 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 ... 5 votes 0 answers 211 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 ... 1 vote 0 answers 93 views ### 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$2m+2$. ... 3 votes 1 answer 364 views ### 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 \...
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 ...