Detecting and Quantifying Isolated Singularities over Discrete Valuation Rings

2025/12/20 by Yotam Svoray, Svoray, Yotam
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · doi:10.48550/arxiv.2512.18506

openalex publication_date 2025/12/20 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28

Abstract

This paper develops a theory of isolated hypersurface singularities in mixed characteristic (0,p), focusing on quotient rings over a Discrete Valuation Ring (DVR). We introduce and study analogues of the classical Tjurina and Milnor numbers for this setting, prove a generalized analogue of the determinacy theorem and the Mather-Yau Theorem for complete Noetherian local rings, and define numerical invariants that provide distinct criteria for detecting isolated singularities in the unramified and ramified cases.

Citations

• ADE Classification of Hypersurface Singularities over Local Rings
• The Splitting Lemma in any Characteristic
• Differential and symbolic powers of ideals
• Singular loci of algebras over ramified discrete valuation rings
• Invariants of Non-Isolated Singularities of Hypersurfaces
• The Tjurina number for Sebastiani-Thom type isolated hypersurface singularities
• The log canonical threshold and rational singularities
• Approximation results of Artin-Tougeron-type for general filtrations and for C^r-equations
• The model theory of Cohen rings
• Singularities in Positive Characteristic
• Milnor and Tjurina numbers for a hypersurface germ with isolated singularity
• Finite determinacy of matrices over local rings.II. Tangent modules to\n the miniversal deformations for group-actions involving the ring\n automorphisms
• Symbolic powers of ideals
• The Theory of Witt Vectors
• A strong version of implicit function theorem
• Mather-Yau Theorem in Positive Characteristic
• Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)
• Constructive characterization of the reduction numbers
• On the structure and ideal theory of complete local rings

Related

• Quantifying singularities with differential operators
• The valuation of the discriminant of a hypersurface
• On the quotient of Milnor and Tjurina numbers for two-dimensional isolated hypersurface singularities
• Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings
• Mather-Yau Theorem in Positive Characteristic