All Questions
Tagged with hilbert-schemes ag.algebraic-geometry
184 questions
4
votes
0
answers
96
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{...
- 215
7
votes
0
answers
188
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 ...
- 171
5
votes
0
answers
285
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 ...
- 479
1
vote
0
answers
101
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$.
...
- 11
3
votes
1
answer
444
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 \...
- 1,677
10
votes
0
answers
405
views
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 ...
- 2,808
5
votes
1
answer
193
views
Multiple of a flat family of subschemes is flat
Let $X$ be a fixed curve (e.g. a Noetherian, projective scheme of dimension 1, of finite type over an algebraically closed field $k$) and let $S$ be an arbitrary parameter scheme over $k$. Let $D \...
- 479
13
votes
1
answer
553
views
On non-representability of certain hom schemes
Let $k$ be an algebraically closed field. It is well-known that the isom-sheaf Isom$(\mathbb{A}^1_k,\mathbb{A}^1_k)$ is not representable by an algebraic space. (To be clear, the functor Isom$(\mathbb{...
- 131
1
vote
0
answers
310
views
Hilbert scheme of Grassmannians
Let $X=\mathbb{G}(k,n)$ be the Grassmannian of $k$-planes in $n$-space. Let $Y\subseteq X$ a subvariety and let $H$ be the connected component of the Hilbert scheme of $X$ that contains $Y$.
Is $H$ ...
- 3,031
1
vote
0
answers
164
views
References for Hilbert schemes over non-Archimedean valuation
Can you suggest me some suitable references to learn the theory of Hilbert polynomials (or related Hilbert schemes) in the non-Archimedean setting?
Thanks.
- 147
4
votes
1
answer
649
views
Exceptional divisor of the Hilbert-Chow morphism of the punctual Hilbert scheme
Let $X$ be a smooth and projective variety of dimension $d>1$. Let $X^{[2]}$ denote the Hilbert scheme of length two subschemes of $X$. Let $X^{(2)}:=X\times X/\mathbb{Z}_2$, where $\mathbb{Z}_2$ ...
- 1,563
3
votes
0
answers
172
views
Do subvarieties naturally map to the hilbert scheme of points?
Let $X$ be a smooth (complex) variety, and $V\subset X$ a reduced, normal subvariety. Fix $k\geq 0$. Then there exists an $n$ such that: for a generic point $v\in V$, we can intersect $V$ with the ...
- 2,824
1
vote
1
answer
290
views
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
2
votes
0
answers
167
views
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
11
votes
1
answer
2k
views
Algebraic cycles, Chow spaces and Hilbert-Chow morphisms
In the sequel, let $S$ be a scheme, and $X$ a locally of finite type algebraic space over $S$.
In his thesis ([R1-R4]), David Rydh introduces, among several others, the notion of relative cycles on $...
user87684
4
votes
0
answers
362
views
Is complete intersection a open or closed property in Hilbert schemes
Fix an integer $N$, $X$ a (smooth) complete intersection subvariety in $\mathbb{P}^N$. Denote by $P$ the Hilbert polynomial of $X$ (as a subvariety in $\mathbb{P}^N$). Consider the Hilbert scheme $\...
- 2,126
6
votes
0
answers
168
views
How does the "todd class operator" commute with Nakajima's q operators on Hilbert schemes of points on surfaces
Let $S$ be a surface, and $S^{[n]}$ the Hilbert scheme of $n$ points on $S$. One defines
$$
\mathbb H = \bigoplus_n H^*(S^{[n]}).
$$
One has Nakajima's operators $\mathfrak q_n(\alpha)$ (with $\alpha \...
- 1,509
11
votes
1
answer
737
views
Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$
What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$?
I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be ...
- 1,537
3
votes
0
answers
121
views
Irreducible but not geometrically irreducible component of Hilbert scheme
If $K$ is a field, is there an irreducible component of the Hilbert scheme ${\rm Hilb}_{\mathbb{P}^r_{K}}$ that is not geometrically irreducible?
- 1,537
2
votes
0
answers
153
views
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
6
votes
0
answers
170
views
A relation of convergence in Hilbert scheme to convergence in sense of currents
Let $\{X_i\}$ be a sequence of closed irreducible $k$-dimensional subvarieties of $\mathbb{C}\mathbb{P}^n$ of degree $d$ (they may be assumed to be smooth if necessary). Assume that this sequence ...
- 21.8k
7
votes
0
answers
368
views
Equivariant Hilbert schemes of points
Let $G$ be a finite subgroup of $\mathrm{SL}(2, \mathbb{C})$ and let $X$ be the quotient surface $X = \mathrm{Spec}(\mathbb{C}[x, y]^{G})$. Denote by $\mathrm{Hilb}^r([X])$ the equivariant Hilbert ...
- 143
14
votes
1
answer
823
views
Counting Hilbert polynomials of projective varieties
EDIT. Fix $n,d,k\in\mathbb{N}$. Let us consider the set $\mathcal{P}_{n,d,k}$ of polynomials $P$ in one variable for which there exits a closed irreducible subvariety $X_P\subset \mathbb{C}\mathbb{P}^...
- 21.8k
0
votes
0
answers
146
views
Limit of a sequence of smooth varieties in Hilbert scheme
Let $\{Z_i\}_{i=1}^\infty$ be a sequence of smooth irreducible $k$-dimensional submanifolds of $\mathbb{C}\mathbb{P}^n$ which converges to a closed subscheme $Z$ in the sense of the Hilbert scheme of $...
- 21.8k
2
votes
0
answers
111
views
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
2
votes
0
answers
161
views
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
5
votes
1
answer
357
views
If $X$ is a degree 3 smooth integral surface in $P^N$, $N > 3$, is it still true that it contains 27 lines?
It is a well known fact that a smooth cubic surface in $P^3$ contains 27 lines. One proof proceeds by moving through the parameter space $U$ of smooth cubics until one reaches an cubic that can be ...
- 1,462
1
vote
0
answers
358
views
A Special Case of Maximal Rank Conjecture
A special case of maximal rank conjecture states that for a general curve $C$ and general points $p_1,\dots ,p_n\in C$ the map
$$Sym^2H^0(K_C-p_1-\dots -p_n)\to H^0(K_C^{\otimes 2}-2p_1-\dots -2p_n)$$
...
- 325
2
votes
0
answers
200
views
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
2
votes
0
answers
74
views
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
2
votes
2
answers
306
views
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
2
votes
0
answers
120
views
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 (...
- 325
5
votes
0
answers
245
views
Is the Chow scheme of 1-cycles the space of Cohen-Macaulay curves?
Let $C\subset X$ be a smooth irreducible curve of genus $g$, embedded in a smooth projective 3-fold $X$. So its homology class $\beta=[C]\in H_2(X)$ is an irreducible class. I want to compare two ...
- 1,534
3
votes
2
answers
758
views
Curves and trisecant lines
We know that rational normal curves and elliptic normal curves have no trisecant lines. For the "next" case, this is still true. That is, a nondegenerate curve of degree $d\geq 5$ and genus $2$ in $\...
- 325
2
votes
1
answer
240
views
Regarding a conjecture Fogarty proposed
In a paper by Fogarty titled "Algebraic Family On An Algebraic Surface,"
he conjectured that $\bf Hilb^n(\mathbb P^N)$ is always variety-- reduced and irreducible.
Is this still a conjecture; any ...
- 21
0
votes
0
answers
160
views
hilbert quot stacks vs schemes
What is the calculation that shows that the Hilbert or Quot functors could be represented by schemes (under various noetherian, (quasi) projectivity hypotheses), and do not require extending to the ...
- 785
8
votes
0
answers
469
views
Hilbert scheme of projectively normal elliptic curves
Consider the Hilbert scheme of degree $n$, genus $1$ curves in $\mathbb P^{n-1}$. It contains the locus of smooth curves embedded by the complete linear system of a degree $n$ divisor. Let $X_n$ be ...
- 148k
2
votes
0
answers
117
views
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
21
votes
1
answer
980
views
$8$-ary operation $(\mathbb{P}^2)^8 \text{ }-\to \mathbb{P}^2$, can we say anything about what this formula would look like?
My friend, who is currently taking an algebraic geometry course from an unnamed prolific poster on MO, told me about the following bonus question on one of his problem sets a few weeks ago.
...
user61522
16
votes
1
answer
1k
views
Reference Request for Hilbert Schemes
I'm a physicist working on Fractional Quantum Hall effect. The mathematical subjects of study are symmetric, translational invariant, homogeneous polynomials on $\mathbb{C}$. Very early in my study I ...
- 613
12
votes
1
answer
2k
views
Hilbert schemes and moduli of ideal sheaves
Let $X$ be a smooth projective variety over $\mathbb{C}$. The Hilbert scheme on $X$ parametrizes quotients $\mathcal{O}_X \to E$ with fixed Hilbert polynomial. Let us fix the Hilbert polynomial to ...
- 783
3
votes
0
answers
362
views
Choosing a group action to do GIT of hypersurfaces
When studying GIT stability of hypersurfaces $d$ of $\mathbb P^n$ we look at the Hilbert Scheme $H=\mathbb P^N$ parametrizing homogeneous polynomials $f_d(x_0,\ldots,x_n)$ of degree $d$. There is ...
- 31
3
votes
1
answer
451
views
Standard techniques on rationally connected varieties
Is there some standard technique or approach to determine when a (irreducible) subvariety of a rationally connected variety is again rationally connected? Any reference/text dealing with this kind of ...
- 2,126
5
votes
1
answer
516
views
Smooth quadric hypersurface, Hilbert scheme is blowup of Grassmannian?
Let $Q \subset \mathbb{P}^n$ be a smooth quadric hypersurface. Where can I find a proof of/can anyone supply a proof of$$\text{Hilb}_{2m + 1}(Q) \cong \text{Bl}_{OG(3, n+1)}G(3, n+1)?$$Can we conclude ...
- 51
1
vote
0
answers
370
views
Hilbert scheme of relative subschemes of lenght 2
Let $\mathfrak X \rightarrow S$ a smooth projective family over the spectrum of a dvr. We know that $(\mathfrak X _{\eta_R})^{[2]}$ and $(\mathfrak X _{p})^{[2]}$ are smooth, where $p$ is the closed ...
- 155
1
vote
0
answers
124
views
Set of smooth curves on the Hilbert scheme is open. H
Let $H = Hilb_{d,g,r}$ be the Hilbert scheme of genus $g$ curves of degree $d$ in proyective space $\mathbb{P}^r$, over an algebraically closed field $k$.
Is it true that the set of points of $H$ ...
- 27
1
vote
0
answers
106
views
Gluing subschemes of fibers of Hilbert Scheme (in mixed characteristic)
Let $\mathfrak{X}\rightarrow Spec(R)$ be a smooth family of smooth projective varieties over a local 1-dimensional ring of mixed characteristic. Suppose that there are non-empty subschemes (locally ...
- 101
1
vote
1
answer
453
views
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
2
votes
1
answer
543
views
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
6
votes
1
answer
338
views
Deformation of curves and closed immersions
Let $\pi:\mathcal{C} \to B$ be a (flat) family of complex projective schemes of pure dimension $1$ with fixed Hilbert polynomial, in particular, for some $n \ge 3$, $\mathcal{C} \hookrightarrow \...
- 2,126