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.

Key Words:

• Hilbert schemes
• 0-dimensional schemes

2000 Mathematics Subject Classification:

• 14B05
• 14C05

ACKNOWLEDGEMENTS

We would like to thank A. Iarrobino and S. Kleiman for pointing out to us that in [1] an example of a singular complete intersection space curve was constructed with reducible Hilbert scheme of points.

The first author is partially supported by MTM2010-15256.

The second author is supported by the research project MTM2009-06964.

Notes

Communicated by L. Ein.