The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces (Q1969184)

\textit{D. Allcock, J. A. Carlsson} and \textit{D. Toledo} [C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 1, 49-54 (1998)] proved that the moduli space of cubic surfaces carries a complex hyperbolic structure. The space is realized as a quotient space of the \(4\)-dimensional ball. It is classically known that it admits a bi-regular action of the Weyl group of type \(E_6\) and can be identified with a Zariski open subset of \({\mathbb C}^4\). Hence, regarding the complex ball lying in the projective space \({\mathbb P}^4\), we have a map from \({\mathbb C}^4\) to \({\mathbb P}^4\) with certain singularities that should have the invariance under the Weyl group and such a map is defined by a differential system of rank 5. This paper gives an explicit form of this system by using the theory of Schwarzian derivatives of several variables.