families of painleve vi equations having a common solution

• DOI: 10.1155/imrn.2005.3727
• Publication Date: 2005-01-01
• Authors: Lubomir Gavrilov; Bassem Ben Hamed
• Keywords: Mathematics - Algebraic Geometry; Mathematics - Classical Analysis and ODEs; 34M55; Classical Analysis and ODEs (math.CA); FOS: Mathematics; 0101 mathematics; 01 natural sciences; Algebraic Geometry (math.AG)

We classify all functions satisfying non-trivial families of PVI equations. It turns out that, up to an Okamoto equivalence, there are exactly four families parameterized by affine planes or lines. Each affine space is generated by points of "geometric origin", associated either to deformations of elliptic surfaces with four singular fibers, or to deformations of three-sheeted covers of the projective line with branching locus consisting of four points.<br/>25 pages, 5 tables and 2 figures. The overall presentation is improved, the solution 1A is added to the main theorem, 3 figures and one table are removed (to save space)<br/>