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Irreducibility of the Hilbert scheme of smooth curves in \(\mathbb {P}^3\) of degree g and genus g

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We denote by \(\mathcal {H}_{d,g,r}\) the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree d and genus g in \(\mathbb {P}^r\). In this note, we show that any non-empty \(\mathcal {H}_{g,g,3}\) is irreducible without any restriction on the genus g. This extends the result obtained earlier by Iliev (Proc Am Math Soc 134:2823–2832, 2006).

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Correspondence to Changho Keem.

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To the memory of our friend and collaborator Ryutaro Horiuchi

The first named author was supported in part by National Research Foundation Grant # 2011-0010298.

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Keem, C., Kim, YH. Irreducibility of the Hilbert scheme of smooth curves in \(\mathbb {P}^3\) of degree g and genus g . Arch. Math. 108, 593–600 (2017). https://doi.org/10.1007/s00013-016-1017-9

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