Pairwise Derived Intersections of Truncated Rulings on a Rational Normal Curve Cone

This work studies pairwise derived intersections of truncated thickenings of rulings on the affine cone over a rational normal curve. It determines minimal graded free resolutions for the corresponding quotient modules and computes their graded Tor modules in every homological degree. The higher Tor modules are independent of whether the two rulings are distinct or coincide, whereas the first Tor module retains information about their geometric support. For distinct rulings, it is described by a finite truncation of a Veronese section module. For a repeated ruling, it has a filtration that separates the generic intersection multiplicity along the ruling from the finite-length contribution concentrated at the cone vertex. The results specialize to symbolic and ordinary powers and provide an associated-graded description of the conormal module arising from self-intersection. A three-factor example shows that the support-independence of the pairwise higher tail does not extend to a support-blind classification of arbitrary thickened derived intersections. The accompanying computation archive independently verifies the finite-dimensional linear-algebra calculations, differential compositions, homology dimensions, and monomial comparisons used in representative cases. 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.