Normalization of Puiseux Hypersurfaces

TL;DR

This paper extends the known normalization results of quasi-ordinary singularities to Puiseux hypersurfaces, establishing a precise correspondence with Hirzebruch-Jung singularities over suitable fields.

Contribution

It proves that the normalization of Puiseux hypersurfaces are exactly Hirzebruch-Jung singularities, generalizing previous results and characterizing their structure.

Findings

Normalization of Puiseux hypersurfaces are Hirzebruch-Jung singularities.

Hirzebruch-Jung singularities are exactly the normalizations of Puiseux hypersurfaces.

Results hold over algebraically closed fields with certain characteristic conditions.

Abstract

It is known that the normalization of a quasi-ordinary complex singularity is a Hirzebruch-Jung, see [Gon00; Pop04; AS05]. We extend this result to Puiseux hypersurfaces. Moreover, we prove that Hirzebruch-Jung singularities are precisely normalizations of Puiseux hypersurfaces. Our result holds over an algebraically closed field whose characteristic does not divide the degree of the polynomial defining the hypersurface. Finally, in the analytic complex case, we conclude that the normalization of an irreducible Puiseux hypersurface is the normalization of a complex analytic quasi-ordinary singularity.