Normalization of Puiseux Hipersurfaces
Fuensanta Aroca, Annel Ayala, Oscar Castañón, Damián Ochoa, Diana Mendez Penagos, Camille Plénat
TL;DR
The paper generalizes the Hirzebruch–Jung normalization property from quasi-ordinary to Puiseux hypersurfaces, showing that the normalization of an irreducible Puiseux hypersurface is either nonsingular or Hirzebruch–Jung, and that every Hirzebruch–Jung singularity arises as such a normalization. It develops a framework of affine semigroups, saturation, and distinguished exponents to connect Puiseux roots with toric normalizations, proving 𝒩(𝕂[[X]][y]/⟨f⟩) ≅ 𝕂[[Ŝ]] for the relevant saturated semigroup Ŝ. In the complex analytic setting, the normalization coincides with the normalization of a complex analytic quasi-ordinary singularity. This work builds a precise toric-analytic bridge between Puiseux hypersurfaces and Hirzebruch–Jung normalizations, advancing the understanding of singularity types and their normalizations.
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.