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

摘要

Denote by Hd,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 Pr. A component of Hd,g,r is rigid in moduli if its image under the natural map pi:Hd,g,r?Mg is a one point set. In this note, we provide a proof of the fact that Hd,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 P3 whose only deformations are given by projective transformations are the twisted cubic curves. In case r >= 4, we also prove the non-existence of a component of Hd,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 Hd,g,3 beyond the range which has been known before.