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Einfacher Zusammenhang der Hilbertschemata von Kurven im komplex-projektiven Raum. (Simple connectedness of the Hilbert schemes of curves in complex projective space). (German) Zbl 0734.14001

Main topic: Let \(H^ d\) be the Hilbert scheme of \(P^ 2_{{\mathbb{C}}}\) of length \(d\) and \(H_{\geq \phi}\) its subscheme of \(p\in H^ d\) with the Hilbert function \(h(p)\geq \phi\), \(\phi\) any function \(\phi: N\to N\). Then \(H_{\geq \phi}\) is simple connected. The proofs needs punctual Hilbert schemes. - A more or less direct corollary of this theorem is: The Hilbert scheme of curves of degree \(d\) and genus \(g\) in the projective space \(P^ 3_{{\mathbb{C}}}\) is simple connected.

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14F45 Topological properties in algebraic geometry
14H10 Families, moduli of curves (algebraic)
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References:

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