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.
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ACKNOWLEDGEMENTS
We would like to thank A. Iarrobino and S. Kleiman for pointing out to us that in [Citation1] 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.