The blow up split sections family

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.