Logarithmic Jacobian ideals, quasi-ordinary hypersurfaces and equisingularity

2025/11/12 by Pérez, Pedro D. González
#Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14M25 #Secondary 32S25

paper · doi:10.48550/arxiv.2511.09126

Abstract

Let (S, 0) ⊂ (ℂ^d+1,0) be an irreducible germ of hypersurface. The germ (S,0) is quasi-ordinary if (S,0) has a finite projection to (ℂ^d,0) which is unramified outside the coordinate hyperplanes. This implies that the normalization of S is a toric singularity. One has also a monomial variety associated to S, which is a toric singularity with the same normalization, and with possibly higher embedding dimension. Since (S,0) is quasi-ordinary, the extension of the Jacobian ideal of S to the local ring of its normalization is a monomial ideal. We describe this monomial ideal by comparing it with the \em logarithmic Jacobian ideals of S and of its associated monomial variety and we give some applications.

Citations

• Nash blowups of toric varieties in prime characteristic
• Overweight deformations of affine toric varieties and local uniformization
• The Frobenius vector of a free affine semigroup
• Motivic Poincaré series, toric singularities and logarithmic jacobian ideals
• Toric Geometry and the Semple-Nash modification
• Analytical invariants of quasi-ordinary hypersurface singularities associated to divisorial valuations
• Decomposition in bunches of the critical locus of a quasi-ordinary map
• Toric embedded resolutions of quasi-ordinary hypersurface singularities
• Equisingularity and simultaneous resolution of singularities
• On the approximate roots of polynomials

Related

• Analytical invariants of quasi-ordinary hypersurface singularities associated to divisorial valuations
• Quasi-ordinary hypersurfaces, multiplier ideals and local tropicalizations
• Toric embedded resolutions of quasi-ordinary hypersurface singularities
• The log canonical threshold and rational singularities
• Toric Geometry and the Semple-Nash modification