Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Twisted stable maps to tame Artin stacks

Authors: Dan Abramovich, Martin Olsson and Angelo Vistoli
Journal: J. Algebraic Geom. 20 (2011), 399-477
Published electronically: September 13, 2010
Erratum: J. Algebraic Geom. 24 (2015), no. 2, 399-400
MathSciNet review: 2786662
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Abstract: We develop the theory of twisted stable maps into a tame Artin stack $ \mathcal{M}$. We show that the stacks $ \mathcal{K}_{g,n}(\mathcal{M})$ of twisted stable maps are algebraic, and proper and quasi-finite over the corresponding stacks $ \mathcal{K}_{g,n}(M)$ of stable maps of the coarse moduli space $ M$ of $ \mathcal{M}$. In the special case where $ \mathcal{M}=\mathcal{B}G$, the classifying stack of a linearly reductive group scheme $ G$, we show that $ \mathcal{K}_{g,n}(\mathcal{B}G)\to\overline{\mathcal{M}}_{g,n}$ is a flat morphism with local complete intersection fibers.

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Additional Information

Dan Abramovich
Affiliation: Department of Mathematics, Brown University, Box 1917, Providence, Rhode Island 02912

Martin Olsson
Affiliation: Department of Mathematics #3840, University of California, Berkeley, California 94720-3840

Angelo Vistoli
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Received by editor(s): April 6, 2008
Received by editor(s) in revised form: October 4, 2009, and March 30, 2010
Published electronically: September 13, 2010
Additional Notes: The first author was supported in part by NSF grants DMS-0301695 and DMS-0603284. The second author was partially supported by NSF grant DMS-0555827 and an Alfred P. Sloan fellowship. The third author was supported in part by the PRIN Project “Geometria sulle varietà algebriche”, financed by MIUR

American Mathematical Society