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.
