All Questions
Tagged with hilbert-schemes schemes
17 questions
1
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0
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108
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L.c.i locus of Hilbert scheme of points on singular varieties
Let $X$ be an algebraic variety over $\mathbb{C}$. What can we say about the l.c.i. locus of $\text{Hilb}^n(X)$?
When $X$ is smooth, it is well-known that the l.c.i. locus of $\text{Hilb}^n(X)$ is ...
- 385
1
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0
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84
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Relation between quot scheme of birational curve
I am very new to algebraic geometry. Currently reading about Hilbert and quot scheme. I want to know more about the structure and properties of Hilbert and quot schemes over curves. My question is the ...
- 613
1
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0
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169
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Étaleness of Isom scheme $\operatorname{Isom}_S(X,Y)$
Let $S$ be a quasi-projective scheme over base field $k$ and $X, Y$ two finite étale schemes over $S$ and assume we are in situation we know that the isom space $\operatorname{Isom}_S(X,Y)$ exists as ...
- 6,028
1
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0
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52
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Symmetric 0-dimensional schemes with generic Hilbert function and Grassmannians
I've came across this problem while thinking about some properties of fat schemes.
Let me give you an explicit (motivating) example:
We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
- 1,343
4
votes
1
answer
359
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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 ...
- 6,028
5
votes
2
answers
461
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Connectedness of Quot schemes
Let $X$ be a connected projective scheme over $\mathbb{C}$ and $E$ a coherent sheaf on $X$. Consider the Quot scheme $\operatorname{Quot}_X(E,P)$ of quotients of $E$ of fixed Hilbert polynomial $P$. ...
- 51
3
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1
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412
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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 ...
- 5,901
2
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1
answer
118
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The weight of a weighted filtration is given (for large $m$) by a polynomial
Let $I$ be an homogeneous ideal of $k[x_0, \dots, x_n]$. Suppose to give integral weights $\lambda_0, \dots, \lambda_n$ to $x_0, \dots, x_n$. We assign a weight to every homogeneous polynomial of ...
- 45
1
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0
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324
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On construction of Hilbert and Quot schemes
I have some questions regarding the strategy of the proof of the existence of Hilbert and Quot schemes (I will focus on the latter since it's more general), as in the book Fundamental Algebraic ...
- 1,906
0
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0
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403
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hypersurface of degree d Hilbert polynomial
I am in trouble to solve part (2) Exercise 1.13 from "Moduli of Curves"
by Harris and Morrison on page 9:
Exercise (1.13)
2) Fix a subscheme $X \subset \mathbb{P}^r$. Show, by taking ...
- 6,028
13
votes
1
answer
553
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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
0
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0
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220
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Use of flattening stratification (from Nitsure's construction of Hilbert and Quot schemes)
I study Nitin Nitsures paper on the Construction of Hilbert and Quot Schemes and not understand the propetry (F) completely:
In previous chapter (Embedding Quot into Grassmanian) it was
proved that ...
- 6,028
2
votes
0
answers
92
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A Subfunctor of Quot-functor compatible with pullbacks
Let $X$ be a smooth projective irreducible algebraic curve over field $k$. For $d,r,k,m >0$ the representable Quot scheme $\mathcal {Quot}_X^{r,d}(\mathcal{O}_X(m)^k)$ is given for
any test scheme $...
- 6,028
2
votes
0
answers
251
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Smoothness of Hilbert scheme of rational normal curves
I'm trying to solve Exercise 1.26 from the book "Moduli of Curves"
by Harris and Morrison on page 14:
Exercise (1.26) Determine the normal bundle to the rational normal
curve $C \subset \...
- 6,028
4
votes
0
answers
96
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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{...
- 215
7
votes
3
answers
652
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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 ...
- 1,811
2
votes
0
answers
117
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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