All Questions
Tagged with hilbert-schemes ag.algebraic-geometry
184 questions
2
votes
1
answer
419
views
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
4
votes
0
answers
410
views
Hilbert vs Chow in nice cases
I'm trying to understand the relationship between the Hilbert schemes and Chow varieties in situations where everything is simple. Suppose that $X$ is a smooth projective variety over $\mathbb C$, ...
user47305
20
votes
3
answers
2k
views
Is there a scheme parametrizing the closed subgroups of an algebraic group?
In the following, let $G=\operatorname{GL}_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more ...
- 4,902
2
votes
0
answers
197
views
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
2
votes
0
answers
251
views
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
1
vote
1
answer
710
views
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
1
vote
0
answers
177
views
Does the canonical morphism commute with the inverse image functor?
I am trying to prove the representability of the Quotient functor.
I have the following problem.
Let $\phi \colon T \to S$ be a morphism of noetherian schemes and let $F$ be a coherent sheaf on $\...
- 31
2
votes
1
answer
288
views
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
2
votes
1
answer
407
views
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
1
vote
0
answers
125
views
Obstruction to Gorenstein Liaisons of space curves
Let $P_1,P$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Denote by $H_{P_1,P}$ the flag Hilbert scheme parametrizing pairs $(C_1 \subset C)$ where $C_1, C$ are of Hilbert polynomials $P_1$ and $...
- 503
3
votes
0
answers
131
views
calculating the Hilbert polynomial of a scheme given its primary decomposition
Given a scheme $X$ in $\mathbb{P}^n$, let $I_X$ be its, saturated, associated ideal. Suppose that a primary decomposition of this ideal is given by
$$
I_X =I_1 \cap \ldots \cap I_2
$$
I was ...
- 103
7
votes
0
answers
293
views
Status of Haiman's conjectures on the Isospectral Hilbert Scheme for dim X>2?
Let $X$ be a variety of arbitrary dimension, let $H$ denote the main component of the Hilbert scheme of points of $X$ (i.e, the closure of locus of reduced subschemes), and let $Z$ be the reduced ...
- 5,958
2
votes
0
answers
104
views
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
1
vote
1
answer
232
views
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
6
votes
0
answers
320
views
A question on infinitesimal deformation (related to intersection theory)
Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then ...
- 833
3
votes
0
answers
226
views
When are Hilbert schemes connected by piecewise smooth curves?
Are there examples of Hilbert scheme $H$ of curves in $\mathbb{P}^3$ such that there exists an irreducible component $L$ of $H$ such that for any two points in $L$, there exist smooth projective ...
- 833
2
votes
0
answers
357
views
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
5
votes
1
answer
164
views
Connectedness of sub-varieties via Hilbert polynomials
Let $X$ be a sub-variety of $\mathbb CP^n$ and let $p_X(k)$ be its Hilbert polynomial. It is well known that some basic invariants of $X$ (such as its dimension) can be read from $p_X(k)$. I am ...
- 14.3k
5
votes
1
answer
437
views
A question on the morphism between Hilbert schemes
Let $L_1,L_2$ be two irreducible component of two different Hilbert schemes parametrizing closed subscheme in $\mathbb{P}^n$ and $\mathbb{P}^{n-1}$, respectively. Denote by $\pi_1: \mathcal{X}_1 \to ...
- 1,981
5
votes
1
answer
831
views
Hilbert scheme of points on a surface
Let $X$ be a complex surface and $X^{[n]}$ be the Hilbert scheme of finite analytic subspaces $Z$ for which $dimH^0(Z,\mathcal{O}_Z)=n$. I have trouble understanding $X^{[n]}$. That's what i've worked ...
- 53
2
votes
1
answer
735
views
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 $\...
- 2,022
3
votes
0
answers
205
views
Projective schemes with a fixed hyperplane section
Let $H$ be a hyperplane in $\mathbb P^n$, and $X \subseteq H$ be a subscheme. Let $CX \subseteq \mathbb P^n$ be the cone on $X$ from a point $p \notin H$.
Let $Hilb_{CX}$ be the Hilbert scheme whose ...
- 27.8k
4
votes
1
answer
533
views
Proving that the Hilbert scheme of points on $\mathbb C^2$ is smooth
On a summer school for undergraduate and graduate students Okounkov gave the following exercise (without hints): Prove that the Hilbert scheme of points on $\mathbb C^2$ is smooth.
Only a definition ...
- 14.3k
3
votes
0
answers
456
views
Deformation of a family of curves in a surface
Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for ...
- 1,040
0
votes
1
answer
195
views
Rational equivalence and infinitesimal deformation of curves
Let $C_1$ and $C_2$ be two rationally equivalent curves in $\mathbb{P}^3$. Is it true that the dimension of $H^0(\mathcal{N}_{C_1|\mathbb{P}^3})$ equal to that of $H^0(\mathcal{N}_{C_2|\mathbb{P}^3})$?...
- 2,022
2
votes
0
answers
108
views
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
1
vote
0
answers
132
views
Functorial property of universal family
Let $P_1, P_2$ and $P_3$ be Hilbert polynomials of projective schemes. Suppose that the flag Hilbert scheme $Hilb_{P_1,P_2}$ is non-empty. Assume further that there exists a closed immersion of the ...
- 1,593
5
votes
1
answer
1k
views
Is projective morphism with projective fiber flat?
Let $X, Y$ be quasi-projective Noetherian schemes and $f:X \to Y$ be a projective surjective morphism. Assume that every fiber of $f$ is isomorphic to a projective space $\mathbb{P}^n$ for a fixed $n$....
- 1,593
9
votes
1
answer
713
views
There are only finitely many varieties up to deformation
Let $h$ be a polynomial. Then results of several authors (including Chow, Grothendieck, Matsusaka, Mumford, Kollar and Viehweg) imply that the moduli space of polarized varieties with Hilbert ...
- 93
4
votes
0
answers
229
views
Non-reduced flag Hilbert schemes
Let $P, Q$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Assume that $pr_2(Hilb_{Q,P})$ is positive dimensional where $pr_2$ is the natural projection map onto the second coordinate and $Hilb_{Q,...
- 1,040
2
votes
1
answer
169
views
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
7
votes
0
answers
421
views
Punctual Hilbert Schemes
Let $H$ be the Hilbert scheme of Artin local rings (quotients of a power series ring $R$ in $e$ variables over $\mathbb{C}$) of length $n$. Consider the set $G\subset H$ of rings $A$ with the property ...
- 6,262
0
votes
0
answers
214
views
Deformation of rational points in a family
Let $\mathcal{X} \to B$ be a family of smooth projective varieties over a field $K$ (possibly finite). Assume that each fiber $\mathcal{X}_b$ of the family has a $K$-rational point. Fix a pair $(p,\...
- 2,022
5
votes
1
answer
965
views
Complete intersection space curves
Fix two positive integers $d, e$ and assume $d>e$. Is it true that a general degree $e$ curve which lies in a complete intersection of a degree $e$ and a smooth degree $d$ surface in $\mathbb{P}^3$ ...
- 2,022
1
vote
1
answer
192
views
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
vote
2
answers
366
views
one "big" Hilbert scheme?
I was recently reading one of the papers of Kapranov on $\mathcal{M}_{0,n}$ and he says that one can see it as a subvariety of the Hilbert scheme parametrizing all subschemes of a certain projective ...
- 3,779
2
votes
0
answers
434
views
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
5
votes
0
answers
325
views
"Reductive Groups and Hilbert Schemes" - Reference
Bezrukavnikov and Ginzburg have unpublished notes, 'Hilbert Schemes and Reductive Groups' (referenced here, for example): does anyone know what became of these notes? Did Bezrukavnikov-Ginzburg ...
- 1,201
3
votes
1
answer
415
views
Singular locus of a Hilbert scheme
Consider the Hilbert scheme $H$ of conics in $\mathbb{P}^3$. It is easy to see that there exists a closed subscheme $H'$ of $H$ parametrizing $2$ lines intersecting at a point. This can be seen as the ...
- 1,040
1
vote
0
answers
122
views
Embedding of curves in surfaces
Let $C_1 \cup C_2$ be a curve in $\mathbb{P}^3$ and $X$ be a smooth degree $d$ surface in $\mathbb{P}^3$ containing them and $d \ge 6$. Further, assume that the minimum degree polynomial in $I(C_1 \...
- 1,040
20
votes
1
answer
822
views
Fuss-Catalan algebras and non-commutative Hilbert schemes
Hello, this is a question regarding Reineke's paper "Cohomology of non-commutative Hilbert schemes", http://arxiv.org/abs/math/0306185, and more precisely the formula on page 4 there (at $n=1$), ...
- 1,363
1
vote
1
answer
640
views
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
1
vote
1
answer
134
views
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
5
votes
2
answers
268
views
Branch locus of a 6:1 cover of the grassmannian G(1,3)
Given a general quartic surface $S$ in $\mathbf{P}^3$, there is a natural 6:1 surjective map
$\phi: Hilb^2(S) \to G(1,3)$ sending $\{P,Q\}$ to the line through them in $\mathbf{P}^3$.
Can you ...
- 453
4
votes
1
answer
470
views
Disconnectedness of Hilbert schemes of projective schemes
Let $Y$ be a projective scheme. The naive definition of a Hilbert scheme of subschemes $X$ of $Y$ would require us to projectively embed $Y$, then ask that $X$ have a fixed Hilbert polynomial $p$.
...
- 27.8k
0
votes
0
answers
401
views
Normal sheaf of non-reduced locally complete intersection space curves
Let $C$ be a non-reduced locally complete intersection curve on a smooth degree $d$ surface in $\mathbb{P}^3$ (for example a non-reduced Cartier divisor). For simplicity we can assume that $d>deg(C)...
- 1,040
2
votes
1
answer
1k
views
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
4
votes
1
answer
226
views
Upper bound on the number of generators of a local complete intersection curve in $\mathbb{P}^3$
Let $C$ be a local complete intersection curve in $\mathbb{P}^3$ (not irreducible or smooth) of degree $e$. Suppose $f_1, f_2$ (and $e_i=\deg(f_i)$) are two of the lowest degree generators of $I(C)$. ...
- 1,040
1
vote
1
answer
178
views
Surjective and injective criteria via Hilbert polynomials
Let $X$ be a projective complex manifold. Is it true that any coherent sheaf $\mathcal{E}$ on $X$ whose Hilbert polynomial is a constant has a surjective morphsim $\mathcal{O}_X\rightarrow \mathcal{M}$...
- 13
3
votes
1
answer
371
views
Hilbert scheme of 2 points on an elliptic curve
The Hilbert scheme of 2 points on an elliptic curve $C$, $Hilb^2(C)$, has a natural structure of ruled surface, given by the map $f:Hilb^2(C) \to C$ such that $f(P,Q)=P+Q$.
What can we say about the ...
- 453