Questions tagged [hilbert-schemes]
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10 questions from the last 365 days
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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
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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
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102
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weak (?) valuative criterion for properness
In the article "On the Kodaira Dimension of the Moduli Space of Curves" by J. Harris and D. Mumford, to prove that
$\overline{H}_{k,b}$ is proper over Spec $\mathbb{C}$, the authors refer to ...
- 560
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158
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Understanding the Hilbert scheme of subvarieties of $\mathbb{CP}^n$
EDIT: migrated to MSE.
I am looking to get a more concrete understanding of the Hilbert scheme of projective subvarieties, specifically over $\mathbb{C}$, and to obtain good references on this subject....
- 1,763
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144
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Curves on the Hilbert scheme of points on surfaces
Suppose $X$ is a smooth projective surface over $\mathbb{C}$ with irregularity $0$ $(q_1(X)=0)$. I want to understand the curves on the Hilbert scheme of $n$-points on $X$.
By the work of Fogarty, we ...
- 335
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128
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Classify all open affine subschemes of a projective variety
Currently I am pondering a question from algebraic geometry that can be stated in very simple terms: Let $X \subseteq P^n_k$ be a projective variety, subscheme of projective $n$-space over an ...
- 708
4
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Nef cone of Hilbert scheme of $n$ points
Suppose $\operatorname{Nef}(X)$ is a rational polyhedron with extremal rays $\{F_i\}_i$. Now, consider the Hilbert scheme of $n$ points $X^{[n]}$ and the embedding $\operatorname{Nef}(X)\subset \...
- 335
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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
1
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224
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Hilbert scheme of points on an arithmetic surface
$\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Spec{Spec}$Let $X$ be a smooth surface over a field $k$. Fogarty proved that the Hilbert scheme of points $\Hilb^n(X)$ is regular. My primary ...
- 573
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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 ...
- 6,018