Reducibility of Punctual Hilbert Schemes of Cone Varieties

Abstract

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ⊆ ℙ N of dimension n and degree d and an integer s 0 such that Hilb s (X) is reducible for all s ≥ s 0. X will be a projective cone in ℙ N over an arbitrary projective variety Y ⊆ ℙ N−1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points.