Combinatorics of Orbit Configuration Spaces

Christin Bibby ; Nir Gadish

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

From a group action on a variety, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this "orbit configuration space" is the complement of an arrangement of subvarieties inside the cartesian product, and we use this structure to study its topology.

We give an abstract combinatorial description of its poset of layers (connected components of intersections from the arrangement) which turns out to be of much independent interest as a generalization of partition and Dowling lattices. The close relationship to these classical posets is then exploited to give explicit cohomological calculations akin to those of (Totaro '96).

Lastly, the wreath product of the group acts naturally. We study the induced action on cohomology using the language of representation stability: considering the sequence of all such arrangements and maps between them, the sequence of representations stabilizes in a precise sense. This is a consequence of combinatorial stability at the level of posets.