  This paper is devoted to two geometric constructions related to the
isomonodromic method. We follow the Drinfeld ideas and develop them in the case
of the curve $X=\mathbb{P}^1\setminus\{a_1,...,a_n\}$. Thus we generalize the
results of Arinkin and Lysenko to the case of arbitrary number $n$ of points.
First, we construct separated Darboux coordinated in terms of the Hecke
correspondences between moduli spaces. In this way we present a geometric
interpretation of the Sklyanin formulas. In the second part of the paper, we
construct Drinfeld's compactification of the initial data space and describe
the compactifying divisor in terms of certain FH-sheaves. Finally, we give a
geometric presentation of the dynamics of the isomonodromic system in terms of
deformations of the compactifying divisor and explain the role of apparent
singularities for Fuchsian equations. To illustrate the results and methods, we
give an example of the simplest isomonodromic system with four marked points
known as the Painlev&#180;e-VI system.
