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

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.