{"id":"hexagon/2610.00231","pdf":"https://hexagonmath.org/pdf/2610.00231v1","url":"https://hexagonmath.org/abs/2610.00231v1","tags":["math.NT","msc:14C22","msc:14J70","msc:11E39","Fermat surfaces","Lean 4","Néron–Severi lattice","discriminant group","lattices","lines on surfaces"],"score":0,"title":"The Watermark and Double Ladder Theorems: the discriminant group of the Néron–Severi lattice of a Fermat surface of prime degree","domain":"math.AG","origin":"ai-generated","source":"hexagon","authors":["Rafael Amichis Luengo"],"reviews":0,"subject":"math.AG","summary":"Let $S_m\\subset\\mathbb{P}^3$ be the complex Fermat surface $x_0^m+x_1^m+x_2^m+x_3^m=0$ of prime degree $m\\ge 5$. Its Néron–Severi lattice $\\mathrm{NS}(S_m)$ is generated by the classes of the $3m^2$ lines on $S_m$ (Schütt–Shioda–van Luijk, Degtyarev). We prove that $|\\mathrm{disc}\\,\\mathrm{NS}(S_m)| = m^{3(m-3)^2}$ (the Watermark Theorem), which answers, for prime degree, the determinant question asked by Shioda in 1987, and we determine the discriminant group itself: $\\mathrm{NS}(S_m)^*/\\mathrm{NS}(S_m)\\cong(\\mathbb{Z}/m)^{3m^2-24m+59}\\times(\\mathbb{Z}/m^2)^{3m-16}$ for $m\\ge 7$, and…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code","Aristotle","Python"],"discussed":0,"published":"2026-10-14","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}