Parameterization of the box variety by theta functions
From the introduction: We consider the graded algebra (the generators have weight one) \(B = \mathbb Q[Z_1, Z_2, Z_3, W_1, W_2, W_3,C]\) with defining relations \[ W_1^2 +W_2^2 = Z_3^2, \quad W_1^2 +W_3^2 = Z_2^2, \quad W_2^2 +W_3^2 = Z_1^2, \quad W_1^2 + W_2^2 + W_3^2 = C^2. \] This is a normal graded algebra. The associated projective variety \(\operatorname{proj}(B)\) is called the box variety. It is absolutely irreducible. We denote its complexification by \(\mathcal B := \operatorname{proj}(B \otimes_{\mathbb Q} C).\) It is a surface that characterizes cuboids. The variables \(W_i\) give the edges of the cuboid, the variables \(Z_i\) the diagonals of the faces and \(C\) the long diagonal. We mention that there is an unsolved problem, raised by Euler, whether the box variety contains non-trivial rational points or not. In this note we describe a parametrization of the box variety (variety of cuboids) by theta functions. This will imply that the box variety is a quotient of the product \(\overline{\mathbb H/\Gamma[8]}\times \overline{\mathbb H/\Gamma[8]}\) of two modular curves of level 8 by a group of order 8 which comes from the diagonal action of \(\Gamma[4]\). Actually this parametrization can be defined over the Gauss number field \(K = \mathbb Q(i)\). We found this parametrization from an observation of D. Testa that the box variety can be embedded into a certain Siegel modular variety which has been described by van Geemen and Nygaard. This parametrization can be used to derive quickly known properties and also some new ones of the box variety. For example, we give in Sect. 2 a modular description of the automorphism group. It can be realized through a subgroup of \(\mathrm{SL}(2, \mathbb Z) \times\mathrm{SL}(2, \mathbb Z)\). In [(*) \textit{M. Stoll} and \textit{D. Testa}, The surface parametrizing cuboids. preprint \url{arxiv:1009.0388}], [(**) \textit{R. van Luijk}, On perfect cuboids. Undergraduate thesis, University Utrecht (2000)] 140 rational and elliptic curves of the minimal model of the box variety which give generators of the Picard group have been described. We describe them in Sect. 3 in a very simple way as certain modular curves. In Sect. 3 we consider smooth curves in the box variety. We prove an estimate that shows how their genus grows with their degree. As a consequence, smooth rational and elliptic curves have a bounded degree. This can be considered as validation of a conjecture made in [(*)] that the 140 curves described in [(**)] exhaust all rational and elliptic curves. This has been proved in [(*)] for degrees \(\le 4\). In the paper [Adv. Stud. Pure Math. 74, 1--10 (2017; Zbl 1388.14108)] of \textit{A. Beauville} the box variety arises as a member of a whole family having the same properties, namely to be complete intersections of 4 quadrics in \(\mathbb P^6\) with an even set of 48 nodes. In this paper Beauville also describes a certain smooth two-fold Galois covering \(X\) of the box variety. It is unramified outside the 48 nodes and it is a minimal surface of general type with \(q = 4\), \(p_g = 7\), \(K^2 = 32\). In Sect. 4 we give a very simple modular description of it. In Sect. 5 a certain involution σ of the box variety \(\mathcal B\) is considered. We use the modular description to realize the quotient \(\mathcal B/\sigma\) as a Kummer variety. In the last section we consider a certain moduli problem which gives the realization of the box variety as fine moduli scheme over \(\mathbb Q(i)\) classifying pairs \((E, F)\) of elliptic curves with level 4 structures and a compatible isomorphism \(E[8] \longrightarrow F[8]\). This is closely related to work of \textit{E. Kani} [Mazur's question and modular diagonal quotient surfaces, preprint (2011)].
scientific article; zbMATH DE number 6694751