Painlevé VI Equations with Algebraic Solutions and Family of Curves

Abstract

In families of Painlevé VI differential equations having common algebraic solutions we classify all the members that come from geometry, i.e., the corresponding linear differential equations that are Picard--Fuchs associated to families of algebraic varieties. In our case, we have one family with zero-dimensional fibers and all others are families of curves. We use the classification of families of elliptic curves with four singular fibers carried out by Herfurtner in 1991 and generalize the results of Doran in 2001 and Ben Hamed and Gavrilov in 2005.