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9 questions
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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
12
votes
2
answers
893
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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 $\mathrm{Hol}(X(\...
- 2,821
1
vote
0
answers
185
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Cycle class/cohomology class of subvarieties in flat families
Let $X$ be a projective variety over $\mathbb C$ and $T$ an irreducible projective $\mathbb C$-scheme.
Let $a,b$ be closed points of $T$.
Suppose we have a flat family $Z\to X\times T\to T$ such that ...
- 111
9
votes
0
answers
387
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Kähler metric on the Hilbert scheme of points on a surface
Question. Let $S$ be a non-singular complex projective surface and let $S^{[n]}$ be its Hilbert scheme of $n$ points. Is there a natural way to associate to a Kähler metric $\omega$ on $S$ a Kähler ...
- 473
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{...
- 4,111
6
votes
0
answers
170
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A relation of convergence in Hilbert scheme to convergence in sense of currents
Let $\{X_i\}$ be a sequence of closed irreducible $k$-dimensional subvarieties of $\mathbb{C}\mathbb{P}^n$ of degree $d$ (they may be assumed to be smooth if necessary). Assume that this sequence ...
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- 1
0
votes
0
answers
146
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Limit of a sequence of smooth varieties in Hilbert scheme
Let $\{Z_i\}_{i=1}^\infty$ be a sequence of smooth irreducible $k$-dimensional submanifolds of $\mathbb{C}\mathbb{P}^n$ which converges to a closed subscheme $Z$ in the sense of the Hilbert scheme of $...
- 21.8k
9
votes
1
answer
713
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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 ...
- 42.9k
1
vote
0
answers
205
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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 ...
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