  The universal scheme of clusters of sections is an adaption of Kleiman's
iterated blow ups (which parametrise clusters of points) to parametrise
clusters of sections. They can also be constructed iteratively, but the
iterative step is not so clear. Defining the blow up split sections family, we
characterise this iterative step. Roughly speaking, it is a morphism that
combines the universal properties of blow ups and universal section families.
It is a generalisation of blow ups, and as such, we show that it exhibits some
sort of birationality. But now, the flattening stratification of a morphism
plays also an important role.
