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Irreducibility of the Hilbert scheme of smooth curves in \(\mathbb {P}^4\) of degree \(g+2\) 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 \(\mathcal {H}_{g+2,g,4}\) is irreducible, generically smooth, and has the expected dimension \(4g+11\) without any restriction on the genus g. Our result augments the irreducibility result obtained earlier by Iliev (Proc Am Math Soc 134:2823–2832, 2006), in which several low genus \(g\le 10\) cases have been left untreated.

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Correspondence to Changho Keem.

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Both authors were supported in part by NRF-South Korea (2017R1D1A1B031763). The authors are grateful to the referee for valuable comments and useful suggestions toward an improvement of the main result of this paper.

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Keem, C., Kim, YH. Irreducibility of the Hilbert scheme of smooth curves in \(\mathbb {P}^4\) of degree \(g+2\) and genus g . Arch. Math. 109, 521–527 (2017). https://doi.org/10.1007/s00013-017-1083-7

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