Wonderful Blow-Ups of Weighted Building Sets and Configuration Spaces of Filtered Manifolds

Abstract

Fulton and MacPherson famously constructed a configuration space that encodes infinitesimal collision data by blowing up the diagonals. We observe that when generalizing their approach to configuration spaces of filtered manifolds (e.g. jet spaces or sub-Riemannian manifolds), these blow-ups have to be modified with weights in order for the collisions to be compatible with higher-order data. In the present article, we provide a general framework for blowing up arrangements of submanifolds that are equipped with a weighting in the sense of Loizides and Meinrenken. We prove in particular smoothness of the blow-up under reasonable assumptions, extending a result of Li to the weighted setting. Our discussion covers both spherical and projective blow-ups, as well as the (restricted) functoriality of the construction. Alongside a self-contained introduction to weightings, we also give a new characterization thereof in terms of their vanishing ideals and prove that cleanly intersecting weightings locally yield a weighting. As our main application, we construct configuration spaces of filtered manifolds, including convenient local models. We also discuss a variation of the construction tailored to certain fiber bundles equipped with a filtration. This is necessary for the special case of jet configuration spaces, which we investigate in a future article.