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07 February 2025
 
  » arxiv » 1605.0297

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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 ;
Date 1 May 2016
AbstractDenote 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 $r=3$. In case $r geq 4$, 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.
Source arXiv, 1605.0297
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