Questions tagged [hilbert-schemes]
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196 questions
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When is the Hom-scheme connected?
Suppose that $A$ and $B$ are two algebras finite over a field $K$ (which may be assumed to be separably closed, if that helps), then we know that the functor $\mathrm{Hom}_K(\mathrm{Spec}(A),\mathrm{...
- 468
2
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1
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407
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Generic vs General property of reducedness in a family of projective schemes
Let $\pi:\mathcal{X} \to B$ be a flat family of projective schemes, $B$ is irreducible. Let $\mathrm{Spec} K$ be a generic point on $B$. Denote by $\mathcal{X}_K$, the pull-back of $\mathcal{X}$. This ...
- 833
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182
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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
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2
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306
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Hilbert scheme of a plane conic union a point
In Alex Lee's undergraduate thesis (2000), it was said that the Hilbert scheme $H_{2m+2}(\mathbb{P}^3)$ has two components $\mathcal{H}',\mathcal{H}''$, where a general point of $\mathcal{H}'$ ...
- 1,537
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288
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Families of curves with "almost-general" moduli
The Brill-Noether theorem says that, if $\rho(d, g, r) := (r + 1)d - rg - r(r + 1) \geq 0$, then there exists a unique component of the Hilbert scheme of curves of degree $d$ and genus $g$ in $\mathbb{...
- 1,832
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1
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258
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Carving out subsheaves of local hom-sheaves of stacks of categories
Recall from my previous question the definition of a local hom-sheaf of a stack of categories. I am interested in stacks of categories such that the underlying stack of groupoids is a moduli stack.
...
- 35.5k
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118
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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 ...
- 45
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420
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Computing Euler Characteristics of Line Bundles on the Hilbert Scheme of n points
Let $S^{[n]}$ be the Hilbert scheme of $n$ points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...
- 1,509
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1
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735
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A question on nested Hilbert scheme
Let $H_1, H_2$ be two Hilbert schemes parametrizing subschemes in $\mathbb{P}^{n_1}, \mathbb{P}^{n_2}$ with Hilbert polynomials $P_1, P_2$, respectively. Given a pair $(Z_1, Z_2)$ of subschemes in $\...
- 2,032
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A question on the Hilbert scheme $I_n(X,\beta)$
Let $X$ be a smooth projective threefold. Let $I_n(X,\beta)$ be the Hilbert scheme parametrizing subschemes $Z \subset X$ with curve class $\beta \in H_2(X,\mathbb{Z})$ and $\chi(\mathcal{O}_Z)=n$. ...
- 21
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Can any local complete intersection subvariety be an intersection of smooth hypersurfaces
Let $Z$ be a local complete intersection subscheme of dimension $m$ in $\mathbb{P}^{2m+1}$. Let $P$ be the Hilbert polynomial of $Z$. Denote by $\operatorname{Hilb}_P$ the Hilbert scheme of local ...
- 1,040
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2
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593
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General degree $d$ surface in $\mathbb{P}^3$
Let $H_{d_1,g_1}, H_{d_2,g_2}$ be two Hilbert schemes of curves in $\mathbb{P}^3$ with degrees $d_1, d_2$ and genus $g_1, g_2$. Denote by $H:=H_{d_1,g_1}\times H_{d_2,g_2}$
where an element in $H$ is ...
- 1,040
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75
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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
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210
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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 ...
- 5,901
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124
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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 ...
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196
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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 ...
- 279
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125
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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 ...
- 5,901
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241
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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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167
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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$ ...
- 23.5k
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92
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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 $...
- 6,028
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275
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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 ...
- 936
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250
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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 \...
- 6,028
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167
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The scheme structure on the Hilbert scheme of an Abel-Jacobi curve
Let $C$ be a smooth curve of genus $g\geq 3$, embedded in its Jacobian $X=\textrm{Jac } C$ via an Abel map. Let $\textrm{Hilb}_1(X)$ be the Hilbert scheme of curves in $X$, and let $[C]\in\textrm{...
- 430
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153
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An analogue of Brill-Noether for hypersurfaces?
Let $d,g,r$ be natural numbers such that $d \geq 1$, $g \geq 2$, $r \geq 3$. Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme classifying subschemes of $\mathbb{P}^r$ with Hilbert polynomial $P(x) = ...
- 1,981
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111
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Subset of a Hilbert scheme consisting of smooth subvarieties
Let $X$ be a smooth projective variety over an algebracally closed field $k$.
(In my case $k=\mathbb{C}, X=\mathbb{P}^n$.)
Let us consider the subset of $k$-points of the Hilbert scheme $Hilb(X)$ ...
- 21.8k
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161
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The non-curvilinear locus in $\textrm{Hilb}^4(\mathbb C^2)$
Let $H_n=\textrm{Hilb}^n(\mathbb C^2)$ be the Hilbert scheme of $n$ points in $\mathbb C^2$ and let $H_n^0\subset H_n$ the punctual Hilbert scheme, parametrizing subschemes entirely supported at the ...
- 430
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200
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Top intersections on the Hilbert scheme of points on a surface
The Picard group of $S^{[n]}$ is generated by the Picard group of $S$ (via a map $L \mapsto L_n$) and $E$, where $E = -\frac{B}{2}$, where $B$ is the exceptional divisor of the Hilbert Chow morphism.
...
- 1,509
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74
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If a subgroup H of a finite group G acts freely on a variety, can the G-Hilbert scheme be computed by iteration?
Let $X$ be a smooth quasi projective variety over $\mathbb{C}$. Let $G$ be a finite abelian group acting via automorphisms on $X$.
Denote by $G$-$\text{Hilb}(X)$ the subscheme of the Hilbert scheme ...
- 1,025
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120
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Transversality of quadrics containing a projective curve
Let $C$ be a curve of genus $g$ and $L$ a $g^r_d$ on it and assume that we are in the range ${r+2\choose 2}>2d-g+1$. If $C$ and $L$ are chosen to be general then by the maximal rank conjecture (...
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117
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representability of some mapping stack
Let $S$ be an Artin stack of finite type.
We assume that it contains a point as an open dense.
Is it always true that the mapping stack:
$Hom^{0}(\mathbb{P}^{1},S)$
which consists of sections ...
- 3,472
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197
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computation with Hilbert scheme of $n$ points on $\mathbb C^2$ [closed]
How can we show that
$$\sum_{n = 0}^\infty q^n \operatorname{char}_T S^n(\mathbb C[x,y])=
\prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$
where $\operatorname{char}_T V$ denotes the character ...
- 533
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251
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Computing Euler Charactistics of Line bundles on Hilbert Schemes of points on Surfaces
Let $S^{[2]}$ be the Hilbert scheme of two points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...
- 1,509
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104
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A basic question on complete intersection liaisons of curves
I am a beginner in the Linkage theory and would like to clarify certain points I am not sure of.
Let $P$ be the Hilbert polynomial of a curve in $\mathbb{P}^3$. Let $L$ be an irreducible component of ...
- 73
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357
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Deformation of complete intersection curves
Let $\mathcal{X} \to B$ be a family of smooth surfaces in $\mathbb{P}^3$. Let $\mathcal{C} \to B$ be a family of curves in $\mathbb{P}^3$. Assume that for all $b \in B$, the fiber $\mathcal{C}_b$ is ...
- 833
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108
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Changing the Hilbert scheme of curves by adding the hyperplane section
Let $X$ be a smooth degree $d$ ($d>4$) surface in $\mathbb{P}^3$.
Let $C$ be a divisor on $X$. Let $D$ be a general element of the linear series $|C+H_X|$ where $H_X$ is the hyperplane section of $...
- 1,040
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434
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Is the universal closed subscheme reduced?
Let $G$ be a finite subgroup of $\text{SL}(2,\mathbb{C})$ and let $Y=\text{GHilb}(\mathbb{C}^2)$ be the minimal resolution of $X=\mathbb{C}^2/G$ where $\text{GHilb}(\mathbb{C}^2)$ is the Nakamura $G$-...
- 1,661
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Hilbert scheme of curves in a degree $d$ surface in $\mathbb{P}^3$
Fix a Hilbert polynomial $P$ of a non-plane curve in $\mathbb{P}^3$. By a curve we mean a reduced scheme of pure dimension $1$ i.e., it can be reducible but is reduced. Suppose that the degree of ...
- 1,040
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443
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Existence of the universal family for the Hilbert scheme of plane curves
Given a finitely generated $k$-algebra $A$ over alg. closed $k$, a family of curves of degree $d$ is defined to be a subscheme $X\subset \mathbb P^2_A$ flat over $A$ whose fibers over closed points of ...
- 2,108
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355
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Boundedness of Hilbert polynomials of hypersurfaces
Let $(X,H)$ be a smooth polarized projective variety of dimension $n$.
If $Y \subset X$ is an irreducible hypersurface then its degree is $H^{n-1} \cdot Y$,
and its Hilbert polynomial is $p_Y(t) = ...
- 8,828
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640
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Hilbert polynomial of $X\times P^1$
Let $X$ be a canonically polarized smooth projective geometrically connected variety over $k$ with Hilbert polynomial $h$.
What is the Hilbert polynomial of $X\times_k \mathbf{P}^1_k$? How does it ...
- 295
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1
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710
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Is Mumford's statement about the representability of some functor wrong?
I am having trouble proving a result in Mumfords book 'Lectures on Curves on an Algebraic surface.
It is a statement about the representability of some functor. It is stated on page 108 and says the ...
- 31
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1
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192
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Is this an embedding of $S^{[2]}$?
The intersection of 3 quadrics in $P^5$ is a K3 surface $S$.
There is a natural map $S^{[2]} \to G(1,5)$ well defined everywhere, because a generic K3 doesn't contain any line and this family is ...
- 453
1
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1
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134
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Upper bound on the dimension of linear series on a smooth hypersurface
Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $d \ge 5$. Let $C \subset X$ be a reduced curve such that $C$ is not a complete intersection curve in $\mathbb{P}^3$. Is it true that ...
- 1,040
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vote
1
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295
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Equivariant homology of Hilb and torus stable curves
The torus equivariant homology of a compact homogeneous variety $G/B$ may be described by using the "skeleton" consisting of torus fixed points and the torus invariant curves that connect them. ...
- 1,335
1
vote
1
answer
290
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Hilbert scheme of points and passing curves
It is well known that through five points on a projective plane you can pass a conic.
Q. What happens when points collide ?
More precisely: if I consider a more simple question of two points and ...
- 399
1
vote
1
answer
453
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Göttsche's formula for non-compact complex surfaces?
Is the Göttsche's formula (Eq (2.1) of this paper) expressing the Poincare polynomial (or the Euler char version) of the Hilbert scheme of points on a projective surface valid for non-compact complex ...
- 11
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1
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232
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Curve of degree $d$ through $2d+1$ points in $\mathbb P^3$
It is known that a Hilbert scheme of degree $d$ curves in $\mathbb P^3$ can have dimension more than $4d$. But, does it imply that for some types of curves there are such a curve through any, say, $2d+...
- 5,063
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1
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174
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$Hilb_{lines}^{x}(X)$ and $Hilb_{lines}^{x}(X_{red})$
Let $X$ be a irreducible closed subscheme of $\mathbb{P}^N_{\mathbb{C}}$,
and $U$ is a nonempty open where $X$ is smooth and moreover for every $x\in U$ and for every line $l\subseteq X$ with $x\in l$ ...
- 1,159
1
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1
answer
226
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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
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1
answer
365
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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