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Irreducibility and components rigid in moduli of the Hilbert scheme of smooth curves

  • Changho Keem
  • Yun-Hwan Kim
  • Angelo Felice Lopez
Article
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Abstract

Denote by \(\mathcal {H}_{d,g,r}\) the Hilbert scheme of smooth curves, that is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree d and genus g in \(\mathbb {P}^r\). A component of \(\mathcal {H}_{d,g,r}\) is rigid in moduli if its image under the natural map \(\pi :\mathcal {H}_{d,g,r} \dashrightarrow \mathcal {M}_{g}\) is a one point set. In this note, we provide a proof of the fact that \(\mathcal {H}_{d,g,r}\) has no components rigid in moduli for \(g > 0\) and \(r=3\), from which it follows that the only smooth projective curves embedded in \(\mathbb {P}^3\) whose only deformations are given by projective transformations are the twisted cubic curves. In case \(r \ge 4\), we also prove the non-existence of a component of \(\mathcal {H}_{d,g,r}\) rigid in moduli in a certain restricted range of d, \(g>0\) and r. In the course of the proofs, we establish the irreducibility of \(\mathcal {H}_{d,g,3}\) beyond the range which has been known before.

Keywords

Hilbert scheme Algebraic curves Linear series Gonality 

Mathematics Subject Classification

Primary 14C05 14C20 

Notes

Acknowledgements

We would like to thank the referee for suggesting that we could add (ii) of Corollary 3.6.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Changho Keem
    • 1
  • Yun-Hwan Kim
    • 1
  • Angelo Felice Lopez
    • 2
  1. 1.Department of MathematicsSeoul National UniversitySeoulSouth Korea
  2. 2.Dipartimento di Matematica e FisicaUniversità di Roma TreRomaItaly

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