  We study the behavior of blocks in flat families of finite-dimensional
algebras. In a general setting we construct a finite directed graph encoding a
stratification of the base scheme according to the block structures of the
fibers. This graph can be explicitly obtained when the central characters of
simple modules of the generic fiber are known. We show that the block structure
of an arbitrary fiber is completely determined by "atomic" block structures
living on the components of a Weil divisor. As a byproduct, we deduce that the
number of blocks of fibers defines a lower semicontinuous function on the base
scheme. We furthermore discuss how to obtain information about the simple
modules in the blocks by generalizing and establishing several properties of
decomposition matrices by Geck and Rouquier.
