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On the Hilbert scheme of smooth curves of degree 15 and genus 14 in \(\mathbb {P}^5\)

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Abstract

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 \(\mathcal {H}_{15,14,5}\) is non empty and reducible with two components of the expected dimension hence generically reduced. We also study the birationality of the moduli map up to projective motion and several key properties such as gonality of a general element as well as specifying smooth elements of each components.

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Notes

  1. Readers are advised not to be confused with another notion of “dual curve" in the sense of Plücker.

  2. The authors are grateful to Angelo Lopez for the example.

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Correspondence to Changho Keem.

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Edoardo Ballico is a member of GNSAGA of INdAM (Italy). Changho Keem was supported in part by National Research Foundation of South Korea (2022R1I1A1A01055306).

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Ballico, E., Keem, C. On the Hilbert scheme of smooth curves of degree 15 and genus 14 in \(\mathbb {P}^5\). Boll Unione Mat Ital (2023). https://doi.org/10.1007/s40574-023-00396-2

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