Thirteen Small Models and Six Resolutions of a Non-Q-Gorenstein Determinantal Blowup

This preprint gives a complete birational classification of the projective normal small models of a specific non-Q-Gorenstein determinantal blowup over an arbitrary field. The blowup is described through its toric fan, singular strata, divisor class group, Picard group, Cartier conditions, and canonical divisor. It is shown that every projective normal small modification of the fixed affine blowup is automatically toric. The resulting relative secondary fan has six chambers and six walls, yielding exactly thirteen small models: six smooth small resolutions, six singular wall models, and the original blowup. The six smooth models form a hexagonal wall-crossing graph. Its edges consist of two ordinary flops and four noncrepant determinantal flips. Explicit smooth common blowups are constructed for all adjacent pairs. The singular wall models are also classified according to their Gorenstein and non-Q-Gorenstein behavior. The accompanying computation files independently verify the fan structure, face lattice, circuit data, divisor-theoretic calculations, chamber decomposition, model count, covering relations, common refinements, and discrepancy data using exact arithmetic. 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.