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On the Hilbert scheme of linearly normal curves in \(\mathbb {P}^4\) of degree \(d = g+1\) 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 article, we show that any non-empty \(\mathcal {H}_{g+1,g,4}\) has only one component whose general element is linearly normal unless \(g=9\). If \(g=9\), we show that \(\mathcal {H}_{g+1,g,4}\) is reducible with two components and a general element of each component is linearly normal. This establishes the validity of a certain modified version of an assertion of Severi regarding the irreducibility of \(\mathcal {H}_{d,g,r}\) for the case \(d=g+1\) and \(r=4\).

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Correspondence to Changho Keem.

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In memory of Professor R.D.M. Accola.

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This paper was prepared for publication when the first named author was enjoying hospitality of the Institute of Mathematics - Academia Sinica (Taiwan) to which he is grateful for the support and the stimulating atmosphere. Both authors were supported in part by National Research Foundation of South Korea (2017R1D1A1B031763).

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Keem, C., Kim, YH. On the Hilbert scheme of linearly normal curves in \(\mathbb {P}^4\) of degree \(d = g+1\) and genus g. Arch. Math. 113, 373–384 (2019). https://doi.org/10.1007/s00013-019-01337-2

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  • DOI: https://doi.org/10.1007/s00013-019-01337-2

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