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Nakajima’s Lectures on Hilbert Schemes of Points on Surfaces begins with a moduli space $\mathcal{M}(r,n)$ of torsion-free sheaves on $\mathbb{P}^2$ with rank $r$, second chern class $n$, trivialized ...
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Let $Y_d,d\geq 2$ be a del Pezzo threefold of Picard rank one and degree $d$. Denote by $\mathrm{Hilb}^{sm}_{d-1}Y_d$ the Hilbert scheme of smooth rational curves of degree $d-1$ on $Y_d$. What is the ...
user41650's user avatar
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Recently I have come across the paper Multigraded Hilbert Schemes by Mark Haiman and Bernd Sturmfels, where they construct a very general object parametrizing homogeneous ideals with fixed Hilbert ...
Carnby 's user avatar
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I have been recently studying the construction of the Hilbert scheme from the book 'Deformations of algebraic schemes' by Sernesi and I am stuck with one of the very first steps. Let me expose the ...
Carnby 's user avatar
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Some questions about reasoning on representability of functor $\underline{\operatorname{Isom}}_S(X,Y)$ for stable curves $X,Y$ over $S$ as given in Definition (1.10) in Deligne's & Mumford's The ...
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The conjecture by Kenyon-Okounkov-Sheffield says that given uniformly chosen lozenge tiling of a hexagon: you can express the random height function of the hexagonal graph (viewing the above picture ...
Pulcinella's user avatar
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for a closed subscheme X of Pn, we have the Hilbert polynomial h(t):= \chi O_X(t). Which polynomials appear as viable Hilbert polynomial? I've listened to hearsay Macauley had a characterization. It ...
ivainsencher's user avatar
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Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{sl}_n\mathbb{C}$, and let $S_n$ denote the Weyl group of $(\mathfrak{sl}_n\mathbb{C}, \mathfrak h)$. I am interested in understanding the ...
Satoshi Nawata's user avatar
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I am currently looking for up-to-date or modern references which discuss the irreducible components of the Hilbert scheme $\text{Hilb}^P (\mathbb{P}^n)$, where $P$ is a fixed Hilbert polynomial. For ...
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Given a Hilbert scheme of curves $\mathrm{Hilb}^P(X)$, with $X$ being smooth irreducible scheme. If $h^0(C, \mathcal{N}_{C/X})$ and $h^1(C, \mathcal{N}_{C/X})$ are constant for all curves $C$ ...
User43029's user avatar
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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 ...
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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 ...
KAK's user avatar
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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 ...
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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....
Paul Cusson's user avatar
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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 ...
Rio's user avatar
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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 ...
Jürgen Böhm's user avatar
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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 \...
Rio's user avatar
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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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$\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Spec{Spec}$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
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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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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 $\mathcal{O}_X$. ...
Yikun Qiao's user avatar
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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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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
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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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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 ...
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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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$\DeclareMathOperator\SL{SL}$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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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
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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
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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
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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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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
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1 answer
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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
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1 answer
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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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$\DeclareMathOperator\Hilb{Hilb}$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
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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
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152 views

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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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
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0 answers
292 views

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
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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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436 views

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
164 views

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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1 answer
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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 ...
Davide's user avatar
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1 answer
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(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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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
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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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0 answers
115 views

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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4 votes
1 answer
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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 $...
user267839's user avatar
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$\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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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
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