JournalsjemsVol. 14, No. 6pp. 1739–1794

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

  • Anca M. Mustaţă

    University College Cork, Ireland
  • Andrei Mustaţă

    University College Cork, Ireland
The structure of a local embedding and Chern classes of weighted blow-ups cover
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Abstract

For a proper local embedding between two Deligne--Mumford stacks YY and XX, we find, under certain mild conditions, a new (possibly non-separated) Deligne--Mumford stack XX', with an etale, surjective and universally closed map to the target XX, 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 YY. Moreover, a natural set of weights on the substacks of XX' allows the construction of a universally closed push-forward, and thus a comparison between the Chow groups of XX' and XX. 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 XX' in the case when XX 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.

Cite this article

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. 1739–1794

DOI 10.4171/JEMS/346