Detecting and Quantifying Isolated Singularities over Discrete Valuation Rings
Yotam Svoray
TL;DR
The paper extends the theory of isolated hypersurface singularities to mixed characteristic settings by working over a DV R and addressing the noninvertibility of the uniformizer pi. It introduces a systematic method to treat pi as a variable, enabling DVR analogues of the Milnor and Tjurina numbers and establishing determinacy results for complete local rings, including a DVR version of the Mather–Yau theorem. Four DVR-specific Tjurina-type invariants (tau_V(f), tau(f, delta), tau^Delta(f), tau^pi(f)) are developed to detect isolated singularities across unramified and ramified cases, with a splitting/Morse-type decomposition and a rank notion guiding the analysis. The framework integrates p-derivation techniques (unramified) and a pi-derivation (ramified) via Hochster–Jacobians and KC-style results, offering new tools for classification and deformation theory in mixed characteristic.
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.