Detecting and Quantifying Isolated Singularities over Discrete Valuation Rings

TL;DR

This paper extends classical singularity theory to mixed characteristic settings over DVRs, introducing new invariants and generalized theorems to detect and analyze isolated hypersurface singularities.

Contribution

It develops a theory of isolated hypersurface singularities in mixed characteristic, introducing analogues of classical invariants and proving generalized determinacy and Mather-Yau theorems.

Findings

Defined new numerical invariants for singularity detection

Proved generalized determinacy theorem in mixed characteristic

Established criteria for isolated singularities in unramified and ramified cases

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.