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.
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We would like to thank the referee for suggesting that we could add (ii) of Corollary 3.6.
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Changho Keem: Supported in part by NRF(South Korea) Grant # 2017R1D1A1B031763.
Angelo Felice Lopez: Supported in part by the MIUR national project “Geometria delle varietà algebriche” PRIN 2010–2011.
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Keem, C., Kim, YH. & Lopez, A.F. Irreducibility and components rigid in moduli of the Hilbert scheme of smooth curves. Math. Z. 292, 1207–1222 (2019). https://doi.org/10.1007/s00209-018-2130-1
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DOI: https://doi.org/10.1007/s00209-018-2130-1