Projective surfaces defined by Appell's hypergeometric systems \(E_4\) and \(E_2\). With an appendix: Differential systems defining non-singular cubic surfaces (Q2772939)

Any surface in the projective 3-space can be described by a system of differential equations in two variables and of rank four. We know only a few explicit examples of such systems where the coefficients are rational functions. Among others, Appell's hypergeometric systems \(E_4\) and \(E^2\) are fundamental. However, a geometric study of such systems, which is promising to give globally defined examples of surfaces would be a progress. The surfaces defined by \(E_4\) are always isothermally asymptotic and the system \(E_4\) provides a concrete example of a 3-parameter family of projectively applicable surfaces. In an appendix the authors present the differential systems defining all non-singular cubic surfaces.