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

Abstract

In this paper we prove that, for anyn≥3, there exist infinitely manyr∈N and for each of them a smooth, connected curveC r in ℙ r such thatC r lies on exactlyn 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.