The blow up split sections family

TL;DR

This paper introduces the blow up split sections family, a new construction generalizing blow ups to parametrise clusters of sections, and characterizes its iterative step using universal properties and flattening stratification.

Contribution

It defines and characterizes the blow up split sections family, extending the concept of blow ups to clusters of sections with a new iterative step.

Findings

The blow up split sections family generalizes blow ups.

It exhibits birationality properties.

The flattening stratification is crucial in its construction.

Abstract

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.