Non-Split Ulrich Defects and a Two-Component Conductor Blowup in a Determinantal Degeneration

Abstract

This paper studies a determinantal degeneration arising from a rank-one matrix cone and analyzes the interaction among a non-split defect module, Ulrich-type structures, conductor geometry, and birational modifications. The defect is described by a globally non-split extension, and its Fitting ideals, dual module, trace, and conductor are determined explicitly. On the distinguished fiber, the associated torsion module admits a pure linear resolution, while two rank-one maximal Cohen–Macaulay modules are related by duality and reflexive multiplication. The conductor blowup is computed through its Rees algebra and normal cone. It is shown to be normal and Cohen–Macaulay, with two irreducible components in the exceptional geometry, but it is not smooth. The paper also compares this blowup with two natural small resolutions, proves that no morphism exists between them in either direction over the original determinantal space, and constructs a common smooth principalization by two successive blowups. The algebraic identities, free resolutions, Rees relations, affine charts, normal-cone decomposition, and local comparison obstructions are accompanied by exact symbolic verification over the integers and in several finite characteristics. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.