Reducibility of Punctual Hilbert Schemes of Cone Varieties

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 ⊆ ℙN of dimension n and degree d and an integer s0 such that Hilbs(X) is reducible for all s ≥ s0. X will be a projective cone in ℙN over an arbitrary projective variety Y ⊆ ℙN-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. © 2013 Copyright Taylor and Francis Group, LLC.