On the Hilbert scheme of curves in higher-dimensional projective space

Abstract

In this paper we prove that, for any $n\ge 3$, there exist infinitely many $r\in \N$ and for each of them a smooth, connected curve $C_r$ in $\P^r$ such that $C_r$ lies on exactly $n$ irreducible components of the Hilbert scheme \hilb(\P^r). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type.Comment: latex, 12 pages, no figure