math.AG — Algebraic Geometry

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

Contributed by Rafael Amichis Luengo

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 $(\mathbb{Z}/5)^{10}\times\mathbb{Z}/25$ for $m=5$ (the Double Ladder Theorem). The transcendental lattice of $S_m$ has the same discriminant group. Both proofs work with the explicit integer intersection matrix $G$ of the lines and use only elementary lattice arithmetic: the action of $G$ on functions pulled back along the $m+1$ pencils of directions of $(\mathbb{Z}/m)^2$, an explicit sublattice of $\ker G$ of index $m^{12}$, and two index computations for the order; an integer identity $GYG=6m^2G$ for the exponent, and a factorization of $G$ modulo $m$ through twelve one-variable "shadows" for the number of cyclic factors of order $m^2$. The lattice statements of both theorems are checked in the proof assistant Lean 4 with the Mathlib library, with no axiom beyond the three standard ones; the identification of $G$ with the intersection matrix of the lines is proved by hand and checked by computer. This work has not been refereed by a human expert.