Painlev\'e VI equations with algebraic solutions and family of curves

TL;DR

This paper classifies Painlevé VI equations with algebraic solutions that originate from geometric families, extending previous classifications of elliptic and algebraic curve families.

Contribution

It identifies all geometric Painlevé VI equations with algebraic solutions, generalizing earlier classifications to include families of curves and zero-dimensional fibers.

Findings

Classification of Painlevé VI equations from geometric families.

Extension of Herfurtner's elliptic curve classification.

Generalization of previous algebraic solution results.

Abstract

In families of Painlev\'e VI differential equations having common algebraic solutions we classify all the members which come from geometry, i.e. the corresponding linear differential equations which 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 done by Herfurtner in 1992 and generalize the results of Doran in 2001 and Ben Hamed and Gavrilov in 2005.