On the Universal Scheme of $r$-relative clusters of a family

TL;DR

This paper extends the concept of point clusters to relative clusters over a base scheme, introducing schemes that parametrize these clusters and analyzing their construction beyond simple blowups.

Contribution

It generalizes $r$-point clusters to $r$-relative clusters over a base scheme and studies their parametrization schemes and complex iterative constructions.

Findings

Defined schemes $Cl_r$ parametrizing $r$-relative clusters

Showed that $Cl_{r+1}$ construction is more complex than a simple blowup

Provided examples illustrating various situations

Abstract

We generalize the concept of $r$-point clusters of a scheme $S$ to $r$-relative clusters of a $B$-scheme $\mathcal{S}$. Define schemes $Cl_r$ that naturally parametrize the $r$-relative clusters which generalize the Kleiman's construction of the iterated blowups by $r$-point clusters. We show that the iterated construction of $Cl_{r+1}$ from $Cl_{r}$ is something more than just to do a blow up of $Cl_r\times_{Cl_{r-1}} Cl_r$ at the diagonal. We show a few simple examples illustrating the situations that may occur.