Smoothness of Concise Border-Rank-Five Forms on Veronese Secant Varieties

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Original abstract

This preprint proves that every point of the fifth Veronese secant variety whose first catalecticant has rank five is smooth. Equivalently, every concise form in five essential variables with border rank five is a smooth point of the ambient fifth secant variety. Consequently, the singular locus is contained in the maximum proper subsecant locus supported on at most four essential variables. This gives an affirmative answer to Question 35 of Furukawa and Han for five secant points. For degrees at least four, the proof combines the classification of forms of minimal border rank with apolarity, smoothable Gorenstein schemes of length five, exact degree-four certificates, and a regularity theorem of Conca and Herzog. Cubic forms are treated using Young flattenings and exact conormal-space certificates, while the quadratic case follows from symmetric determinantal geometry. The accompanying computation archive contains exact integer certificates and verification programs. The programs reconstruct the relevant matrices from the recorded normal forms and verify the required rank, kernel, determinant, coefficient, and consistency conditions 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.

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