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0
votes
0answers
110 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 ...
8
votes
0answers
140 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 ...
2
votes
0answers
88 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 ...
29
votes
1answer
606 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. ...
9
votes
4answers
313 views

Moduli spaces in applied mathematics and condensed matter physics?

In this MO question it is stated that there is a relation between some aspects of condensed matter physics (namely the fractional quantum Hall effect) and the algebraic geometry of Hilbert schemes. ...
12
votes
1answer
392 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 ...
8
votes
1answer
586 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 ...
3
votes
0answers
95 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 ...
3
votes
1answer
153 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 ...
7
votes
1answer
210 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 ...
0
votes
0answers
100 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 ...
1
vote
0answers
99 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 $H$ ...
1
vote
0answers
70 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 ...
1
vote
1answer
167 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 ...
2
votes
1answer
227 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 ...
4
votes
1answer
194 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
votes
1answer
186 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 ...
3
votes
0answers
307 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 $\mathbb C$, ...
12
votes
2answers
938 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 ...
2
votes
0answers
167 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 ...
2
votes
0answers
111 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
vote
1answer
586 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 ...
1
vote
0answers
110 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 ...
1
vote
1answer
170 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 ...
2
votes
1answer
135 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 ...
1
vote
0answers
88 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 ...
2
votes
0answers
72 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 ...
5
votes
0answers
156 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 ...
2
votes
0answers
75 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 ...
0
votes
1answer
166 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, ...
6
votes
0answers
285 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 ...
2
votes
0answers
157 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 ...
2
votes
0answers
158 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 ...
5
votes
1answer
125 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 ...
5
votes
1answer
255 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 ...
4
votes
1answer
418 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 ...
1
vote
1answer
304 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 ...
3
votes
0answers
169 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 ...
4
votes
1answer
326 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 ...
2
votes
0answers
287 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 ...
0
votes
1answer
135 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 ...
2
votes
0answers
76 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
vote
0answers
106 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 ...
4
votes
1answer
567 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 ...
9
votes
1answer
568 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 ...
3
votes
0answers
152 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 ...
2
votes
1answer
135 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 $\chi(\mathcal{O}_Z)=n$. ...
2
votes
0answers
184 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 ...
0
votes
0answers
159 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 ...
4
votes
1answer
443 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 $\mathbb{P}^3$ ...