Singularities of the isospectral Hilbert scheme

Abstract

We study the singularities of the isospectral Hilbert scheme Bⁿ of n points over a smooth algebraic surface and we prove that they are canonical if $n\le 5$, log-canonical if $n\le 7$ and not log-canonical if $n\ge 9$. We describe as well two explicit log-resolutions of B³, one crepant and the other ${\mathfrak{S}}_3$-equivariant.

Keywords:

• Isospectral Hilbert scheme of points
• canonical singularities
• log-canonical thresholds
• crepant resolution

2010 Mathematics Subject Classification:

• 14B05
• 14E15
• 14C05

Acknowledgments

I would like to sincerely thank Lei Song for inviting me to University of Kansas, for his interest in my work and for communicating the results mentioned in Section 2.4. I would also like to thank the referee for the useful suggestions and comments which improved the exposition of the paper.

Additional information

Funding

This work is partially supported by CNPq, grant 307795/2012-8.