Full-Secant Singularities of Veronese Varieties with Reduced Singleton Border VSP

Abstract

This paper gives a negative answer to Question 35 of Furukawa and Han by constructing an explicit family of singular points on full secant varieties of Veronese varieties. The construction works for every border rank of at least six, every compatible ambient dimension, and every degree of at least three. These points lie outside the preceding secant variety and outside all ordinary and partial subsecant loci considered in the question. The tangent-space dimension is determined exactly for the entire family. Each form has a unique saturated apolar scheme of minimal length, and its border variety of sums of powers is a reduced singleton. The associated abstract border-secant map is locally an isomorphism. Consequently, the singularity is inherited from the unique smoothable apolar scheme rather than arising from multiple border decompositions or from an infinitesimally nonreduced decomposition space. The paper also exhibits an explicit projective linear space contained in the singular locus and describes the drop in border rank along its boundary. The accompanying computation archive provides exact symbolic checks for the principal identities and rank calculations used in the proofs. 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.