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

Abstract

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 $X^{\prime }$, with an etale, 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 etale, surjective and universally closed maps to $Y$. Moreover, a natural set of weights on the substacks of $X^{\prime }$ allows the construction of a universally closed push-forward, and thus a comparison between the Chow groups of $X^{\prime }$ 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 $X^{\prime }$ 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.