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

For a local embedding of Deligne-Mumford stacks g : Y ! X, we find an ´etale cover of X by a (non-separated) Deligne-Mumford stack Xsuch that the fiber product Y ' = g(Y ) ×X Xis a finite union ofetale covers Y ' i of Y , and such that the maps Y ' ! Y and X ' ! X are universally closed. Moreover, a natural set of weights on the substacks of Xallows the construction of a universally closed push-forward, and thus a comparison between the Chow rings of Xand X. In the case when X is the moduli space of stable maps or one of its intermediate spaces, the Chow ring of X ' has been introduced and calculated in (MM1) as an extended Chow ring of a network. Here we identify the corresponding stack Xand its moduli problem in terms of stable maps with marked components. We apply the construction above to computing the Chern classes of a weighted blow-up along a regular local embedding via deformation to a weighted normal cone and localization. We apply these calculations to find the Chern classes of the stable map spaces.