The Bolza curve and some orbifold ball quotient surfaces (Q1980019)

The Bolza curve refered to in the title is a genus 2 curve \(C\) with maximal automorphisms group \(\mathrm{Aut}(C)\simeq\mathrm{GL}_2(\mathbb{F}_3)\) to be denoted by \(G_{48}\) in the sequel. Its Jacobian is thus an Abelian surface \(A\) endowed with an action of \(G_{48}\) and this article is concerned with the study of the quotient \(A/G_{48}\). It is proven that the latter is isomorphic to the weighted projective space \(\mathbb{P}(1,3,8)\) (Theorem~A). It is interesting to have such a geometrical description: indeed \textit{M. Deraux} [Comment. Math. Helv. 93, No. 3, 533--554 (2018; Zbl 1402.22011)] proved that some bimeromorphic models of \(A/G_{48}\) carry orbifold structures in codimension 1 that correspond to ball quotients \(\mathbb{B}^2/\Gamma\) for some lattices \(\Gamma<\mathrm{PU}(1,2)\). This article (combined with Dearux's results) gives thus a complete description of such orbifold ball quotients, description that is fairly hard to obtain in general. To be precise, with the identification \(A/G_{48}\simeq\mathbb{P}(1,3,8)\) at hands, the authors introduce then 2 birational maps from \(\mathbb{P}(1,3,8)\) to \(\mathbb{P}^2\) and \(\mathbb{P}^1\times\mathbb{P}^1\) and they carefully study the images of \(M_{48}\) under these 2 maps, where \(M_{48}\) is the codimension 1 component of the branching locus of the projection \(A\to A/G_{48}\). The curve obtained in \(\mathbb{P}^2\) is denoted \(M_{\mathbb{P}^2}\) (resp. \(M_{\mathbb{P}^1\times\mathbb{P}^1}\subset\mathbb{P}^1\times\mathbb{P}^1\)). They completely compute these 2 curves and, as a byproduct, they prove that \(M_{\mathbb{P}^2}\) is the unique quartic plane curve satisfying some conditions on singular points and tangent lines at flex points (see Theorem 14 for a precise statement). The authors finally use this to get explicit orbifold ball quotients as surfaces with curves configurations. Parts of computations are carried over using Magma (the full Magma code is available as an ancillary file of the arXiv version \url{arXiv:1904.00793v4}).