Polydiagonal compactification of configuration spaces

A smooth compactification $X\!\left \langle n\right \rangle$ of the configuration space of $n$ distinct labeled points in a smooth algebraic variety $X$ is constructed by a natural sequence of blowups, with the full symmetry of the permutation group ${\mathbb S}_n$ manifest at each stage. The strata of the normal crossing divisor at infinity are labeled by leveled trees and their structure is studied. This is the maximal wonderful compactification in the sense of De Concini–Procesi, and it has a strata-compatible surjection onto the Fulton–MacPherson compactification. The degenerate configurations added in the compactification are geometrically described by polyscreens similar to the screens of Fulton and MacPherson.

In characteristic 0, isotropy subgroups of the action of ${\mathbb S}_n$ on $X\!\left \langle n\right \rangle$ are abelian, thus $X\!\left \langle n\right \rangle$ may be a step toward an explicit resolution of singularities of the symmetric products $X^n\!/{\mathbb S}_n$.