The Watermark and Double Ladder Theorems: the discriminant group of the Néron–Severi lattice of a Fermat surface of prime degree
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…
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