Toward Finding New Differential Equations

Abstract

In Chapter 10, we showed that for each line arrangement A ⊂ P₂ given in § 10.3, there is a Fuchsian differential equation defined on P₂ of rank 3 with regular singularities along A (the uniformizing differential equation of the orbifold). For complete quadrilateral arrangement, the equation is the Appell’s system F₁, which was studied in § 10.5. In this chapter, we find differential equations for the arrangements given in Examples (2), ..., (8). So this is a continuation of § 10.5. On the other hand, as we mentioned in § 9.1, it can be considered as a method for solving the Riemann problem in a very restricted case.