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7
votes
3answers
601 views

Families of ideal sheaves: What's the correct definition?

I'm looking at Bridgeland's paper "Flops and Derived categories" and I got confused on what he meant by a family of ideal sheaves. Let $Y$ be a scheme, and let $S$ be another scheme. A family of ...
2
votes
1answer
276 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 $C \subset \mathbb{P}^3$ ...
0
votes
0answers
137 views

Maximum number of generators of a curve in $\mathbb{P}^3$

Let $H_{d,g}$ denote the Hilbert scheme of curves of degree $d$ and genus $g$ locally of complete intersection in $\mathbb{P}^3$. Given a curve $C \in H_{d,g}$, denote by $S(C)$ a minimal set of ...
1
vote
1answer
177 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 $d$ ...
2
votes
2answers
270 views

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 ...
5
votes
3answers
357 views

Irreducible “family” of relative effective divisors of a smooth morphism

Let $\pi:X\rightarrow Y$ be a smooth proper (assume projective if needed) morphism of schemes with $Y$ locally noetherian, and let $Z\subset X$ be an irreducible integral closed subscheme containing ...
1
vote
2answers
310 views

Any irreducible component of the HIlbert scheme contains an irreducible element

Consider $Hilb_{d,g}$ the Hilbert scheme of curves in $\mathbb{P}^3$ of degree $d$ and genus $g$. Is it true that if $L$ is an irreducible component of $Hilb_{d,g}$ then there exists a curve $C \in L$ ...
5
votes
1answer
624 views

Examples of nice reduced singularities on Hilbert schemes--Edited

In his "Murphy's Law" paper, Vakil showed that every "singularity type" (with a precise meaning) occurs on certain Hilbert schemes; for instance, the Hilbert scheme of nonsingular curves in projective ...
1
vote
0answers
148 views

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 ...
0
votes
1answer
340 views

Irreducible components of the Hilbert scheme

Let $P_1, P_2, Q$ denote the Hilbert scheme of a plane conic in $\mathbb{P}^3$, a quartic and a degree $d$ surface in $\mathbb{P}^3$. Then there is a natural inclusion map $i$ from Hilbert flag scheme ...
2
votes
1answer
194 views

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. ...
1
vote
0answers
153 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 $C$ ...
1
vote
0answers
182 views

Space of sections

If S is a noetherian scheme and π : Z → X a morphism of S-schemes, where X is proper over S and Z is quasi-projective over S, then the set-valued contravariant functor $\Pi_{Z/X/S}$ on locally ...
1
vote
1answer
167 views

$Hilb_{lines}^{x}(X)$ 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 $x\in l$ ...
2
votes
1answer
214 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 ...
5
votes
0answers
420 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 ...
6
votes
2answers
736 views

Relationship between Hilbert schemes and deformation spaces

Hi, I'm just starting to learn about deformation theory (via Hartshorne's Deformation theory, as well as Fantechi's section of FGA explained), and I feel like I'm confused about fundamental concepts. ...
3
votes
0answers
185 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 ...
1
vote
1answer
189 views

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. ...
9
votes
3answers
679 views

Reference for combinatorics of cell decomposition of the Hilbert scheme of points in the plane

It is known from either Morse theory or Bialynicki-Birula decomposition that the fixed points of a ${\mathbb{C}}^*$ action on a smooth algebraic variety over $\mathbb{C}$ determine a cell ...
1
vote
0answers
242 views

Regularity and limits of smooth rational curves.

Fix integers $2 < d \leq n$. Suppose that $T$ is a smooth complex curve with marked point $0 \in T$, and $X$ is a closed subscheme of $\mathbb{P}^n_T$, flat over $T$ such that each fiber has ...
1
vote
0answers
203 views

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) = ...
10
votes
2answers
431 views

Why is Maps(X,Y) an open subfunctor of Hilb(X x Y)?

Let $X$ and $Y$ be projective schemes. Then we can define the mapping scheme between them, $\rm{Maps}(X,Y)$ as follows: To any map $f:X\rightarrow Y$ we consider the graph $\Gamma_f$ as a closed ...
2
votes
1answer
240 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 $f \equiv 1$, ...
9
votes
2answers
568 views

Are spaces of holomorphic maps manifolds?

Hello, Let $X$ and $Y$ be two smooth (probably projective) algebraic varieties defined over $\mathbf{C}$. What is known in general about the (topological) space of holomorphic maps ...
7
votes
3answers
903 views

Reference request: is the punctual Hilbert scheme irreducible?

The punctual Hilbert scheme in dimension $d$ parameterizes ideals $I$ of codimension $n$ in $k[x_1,\dots, x_d]$ which are contained in some power of the ideal $(x_1,\dots, x_d)$. In other words, it is ...
16
votes
2answers
952 views

Why do flag manifolds, in the P(V_rho) embedding, look like products of P^1s?

Bert Kostant mentioned an odd fact to me some time ago. As usual (with such statements), fix a complex, connected, reductive) Lie group $G$, with maximal torus $T$, and Weyl vector $\rho$ equal to ...
7
votes
1answer
286 views

Is there a way to check if a relative Hilbert Scheme is reduced?

More specifically, suppose I have a rational curve on a complete intersection, and I know that the relative Hilbert Scheme is not smooth at the point corresponding to my rational curve. Is there any ...
4
votes
1answer
315 views

What are the “special” strata of Sym^n(C^2)?

The affine variety $Sym^n(\mathbb{C}^2)$ has a natural quantization as a spherical rational Cherednik algebra. Thus, any primitive ideal of the rational Cherednik algebra has an corresponding ideal ...
1
vote
1answer
230 views

Request for info on the space of commuting matrices preserving a flag.

Fix a flag of subspaces V1 in V2 in V3, etc. all in Cn. Consider the space of pairs of commuting linear transformations A and B such that: A preserves the flag (i.e. A(Vi) is in Vi), and B strictly ...
6
votes
4answers
1k views

When are Hilbert schemes smooth?

I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a ...