Using the Gauss-Manin connection (Picard-Fuchs differential equation) and a result of Malgrange, a special class of algebraic solutions to isomonodromic deformation equations, the {\em geometric isomonodromic deformations}, is defined from ``families of families'' of algebraic varieties. Geometric isomonodromic deformations arise naturally from combinatorial strata in the moduli spaces of elliptic surfaces over ${\mathbb{P}}^1$. The complete list of geometric solutions to the Painlev\'{e} VI equation arising in this way is determined. Motivated by this construction, we define another class of algebraic isomonodromic deformations whose monodromy preserving families arise by ``pullback'' from (rigid) local systems. Using explicit methods from the theory of Hurwitz spaces, all such algebraic Painlev\'{e} VI solutions coming from arithmetic triangle groups are classified.