Polydiagonal compactification of configuration spaces

Abstract

A smooth compactification X<n> 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 S_n manifest at each stage of the construction. The strata of the normal crossing divisor at infinity are labeled by trees with levels and their structure is studied. This is the maximal wonderful compactification in the sense of DeConcini-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 screens of Fulton and MacPherson. In characteristic 0, isotropy subgroups of the action of S_n on X<n> are abelian, thus X<n> may be a step toward an explicit resolution of singularities of the symmetric products X^n/S_n.