The structure of a local embedding and Chern classes of weighted blow-ups

For a proper local embedding between two Deligne-Mumford stacks Y and X, we find, under certain mild conditions, a new (possibly non-separated) Deligne-Mumford stack X′, with an étale, surjective and universally closed map to the target X, and whose fiber product with the image of the local embedding is a finite union of stacks with corresponding étale, surjective and universally closed maps to Y. Moreover, a natural set of weights on the substacks of X′ allows the construction of a universally closed push-forward, and thus a comparison between the Chow groups of X′ and X. We apply the construction above to the computation of the Chern classes of a weighted blow-up along a regular local embedding via deformation to a weighted normal cone and localization. We describe the stack X′ in the case when X is the moduli space of stable maps with local embeddings at the boundary. We apply the methods above to find the Chern classes of the stable map spaces.