Questions tagged [hilbert-schemes]

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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 ...
Malkoun's user avatar
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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 ...
Drew's user avatar
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155 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 $\...
var's user avatar
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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 ...
Hans's user avatar
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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 ...
Raffaele C's user avatar
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230 views

Is a Procesi bundle equipped with a hyperholomorphic connection?

Haiman has constructed in the paper the unusual tautological bundle $P$, called Procesi bundle, of rank $n!$ over the Hilbert schemes of points on the affine plane in the following way. Let $H _ { n }...
Satoshi  Nawata's user avatar
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244 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 ...
Brenin's user avatar
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726 views

nth symmetric power of a Riemann surface and its Jacobian

Let $C$ be a Riemann surface of genus $g$, $Sym^{n}C$ be the nth symmetric power of $C$ with $n\geq 2g$, and $JC$ denote the Jacobian of $C$. Question: Is it generally true that $Sym^{n}C\cong JC\...
YOURS's user avatar
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4 votes
1 answer
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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 $p$. ...
Allen Knutson's user avatar
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535 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 ...
Ron's user avatar
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irreducibility punctual Hilbert scheme of relative subschemes of length $2$

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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1 answer
483 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 ...
aglearner's user avatar
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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 ...
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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 $\mathbb{Z}_2$ ...
Rex's user avatar
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1 answer
220 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 $I(C)$. ...
Naga Venkata's user avatar
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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 $Z$ ...
Learner2000's user avatar
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0 answers
123 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....
Yellow Pig's user avatar
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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{...
BenediktK's user avatar
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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 $\...
Ron's user avatar
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4 votes
0 answers
407 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 $\mathbb C$, ...
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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,...
Naga Venkata's user avatar
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321 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 ...
George Melvin's user avatar
3 votes
1 answer
497 views

linear system of non-reduced divisor and associated reduced divisors

Let $X$ be a smooth degree $d$ $(d \ge 5)$ surface in $\mathbb{P}^3$. Let $D$ be an effective Cartier divisor (hence locally of complete intersection) on $X$ and $D_{red}$ the associated reduced ...
Naga Venkata's user avatar
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3 votes
1 answer
401 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 ...
Naga Venkata's user avatar
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3 votes
1 answer
361 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 ...
sqrt2sqrt2's user avatar
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1 answer
253 views

On the place where $\mathrm{Hilb}_{lines}^{x}(X)$ is smooth.

Let $X\subset \mathbb{P}_{\mathbb{C}}^N$ be irreducible generically smooth closed subscheme and let $\mathrm{Hilb}_{lines}^{x}(X)$ denote the Hilbert scheme of lines contained in $X$ and passing ...
gio's user avatar
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3 votes
2 answers
704 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 $\...
Irfan Kadikoylu's user avatar
3 votes
1 answer
433 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 ...
Ron's user avatar
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3 votes
1 answer
352 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 ...
user127776's user avatar
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3 votes
1 answer
378 views

Deformation of space curves to union of lines

Does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)? Furthermore, given a curve $C \subset \mathbb{P}^3$ ...
Naga Venkata's user avatar
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3 votes
1 answer
358 views

When is the natural projection of the HIlbert flag scheme a flat morphism

Let ${Hilb_{P,Q}}_{red}$ be the reduced scheme associated to the Hilbert flag scheme parametrizing all pairs $(C,X)$ with $C \subset X \subset \mathbb{P}^3$, where $C$ is a curve and $X$ a degree $d$ ...
Naga Venkata's user avatar
  • 1,010
3 votes
1 answer
411 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 \...
Filip's user avatar
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3 votes
0 answers
190 views

Virtual fundamental class of punctual Hilbert scheme of points

$\DeclareMathOperator\Hilb{Hilb}$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
3 votes
0 answers
198 views

Deformations of genus g curves to 'non-reduced rational curve'

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
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 ...
jacob's user avatar
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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?
DCT's user avatar
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3 votes
0 answers
331 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 ...
John's user avatar
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3 votes
0 answers
126 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 ...
NotNow's user avatar
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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 ...
user46578's user avatar
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3 votes
0 answers
202 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 ...
Allen Knutson's user avatar
3 votes
0 answers
448 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 ...
Naga Venkata's user avatar
  • 1,010
3 votes
1 answer
357 views

Upper bound on the dimension of the Hilbert scheme of space cuves

Denote by $H_{P,Q}$ the flag Hilbert scheme parametrizing a pair $(C,X)$ such that $X$ is a degree $d$ surface in $\mathbb{P}^3$ with Hilbert polynomial $Q$ and $C \subset X$ is a curve with Hilbert ...
Naga Venkata's user avatar
  • 1,010
3 votes
0 answers
309 views

Hilbert function of a Hilbert scheme

Given a Hilbert scheme $H$ of curves in $\mathbb{P}^3$ satisfying certain Hilbert polynomial, is there any way of understanding the degree or arithmetic genus of an irreducible component of the ...
Naga Venkata's user avatar
  • 1,010
3 votes
0 answers
249 views

Smooth curve in the Hilbert flag scheme

Let $d$ be an integer greater than $0$. Let $P_2$ be the Hilbert polynomial of a degree $d$ surface in $\mathbb{P}^3$. Recall, the Hilbert flag scheme $\mathrm{Hilb}_{P_1,P_2}$ parametrizes curves $C$ ...
Naga Venkata's user avatar
  • 1,010
3 votes
0 answers
287 views

Polarizations on $M_{0,n}$ from Kapranov's quotient constructions

In Kapranov's marvelous paper Chow quotients of Grassmannian I, he proves that $\overline{M}_{0,n}$ is isomorphic to both the Hilbert quotient and Chow quotient $(\mathbb{P}^1)^n//\text{SL}_2$. These ...
Noah Giansiracusa's user avatar
2 votes
1 answer
273 views

What is the family derived from the absolute Frobenius on the Hilbert scheme?

Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, ...
Allen Knutson's user avatar
2 votes
1 answer
315 views

tangent bundle of Hilbert schemes of points on a projective surface

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
2 votes
1 answer
227 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 ...
Algeometry's user avatar
2 votes
1 answer
528 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{...
Jizhan Hong's user avatar
2 votes
1 answer
385 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 ...
user46578's user avatar
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