# Questions tagged [hilbert-schemes]

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### What can one say about a subscheme of a Hilbert scheme, which is covered by lines?

k= complex numbers, X/k closed subscheme of a Grassmannian, which is Plücker embedded in a projective space. a)X is simply connected b) the first Chow group (rational coefficients)of X is generated by ...
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### Open subfunctor of Quot Functor induced by open immersion

Let $f: X \rightarrow S$ be a morphism of noetherian schemes and $\mathfrak{Q}uot_{\mathcal{E}/X/S}$ be the functor parametrizing families of quotients of $\mathcal{E}$ in the category of locally ...
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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 ...
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### Uniqueness of the scheme structure for a given Hilbert polynomial

If we have two lines in $P^3$ which are skewed, then we can take the union of those lines as a subscheme of $P^3$ in order to obtain a subscheme of $P^3$ with a Hilbert Polynomial given by $2m+2$. ...
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### Hilbert scheme of Grassmannians

Let $X=\mathbb{G}(k,n)$ be the Grassmannian of $k$-planes in $n$-space. Let $Y\subseteq X$ a subvariety and let $H$ be the connected component of the Hilbert scheme of $X$ that contains $Y$. Is $H$ ...
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### References for Hilbert schemes over non-Archimedean valuation

Can you suggest me some suitable references to learn the theory of Hilbert polynomials (or related Hilbert schemes) in the non-Archimedean setting? Thanks.
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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 $\mathbb{Z}_2$ ...
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### 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 ...
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### Hilbert scheme of points and passing curves

It is well known that through five points on a projective plane you can pass a conic. Q. What happens when points collide ? More precisely: if I consider a more simple question of two points and ...
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### Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$

What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$? I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be ...
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### Constructing embedded families of curves with general moduli

Is there a known way to construct flat families of smooth curves $\mathcal{C}/\mathcal{B}$ which are fiberwise embedded in a family of projective varieties $\mathcal{X}/\mathcal{B}$ and which have ...
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### 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?