For a proper local embedding between two Deligne--Mumford stacks and , we find, under certain mild conditions, a new (possibly non-separated) Deligne--Mumford stack , with an etale, surjective and universally closed map to the target , and whose fiber product with the image of the local embedding is a finite union of stacks with corresponding etale, surjective and universally closed maps to . Moreover, a natural set of weights on the substacks of allows the construction of a universally closed push-forward, and thus a comparison between the Chow groups of and . 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 in the case when 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.
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Anca M. Mustaţă, Andrei Mustaţă, The structure of a local embedding and Chern classes of weighted blow-ups. J. Eur. Math. Soc. 14 (2012), no. 6, pp. 1739–1794DOI 10.4171/JEMS/346