Twisted stable maps to tame Artin stacks

Abramovich, Dan; Olsson, Martin; Vistoli, Angelo

doi:10.1090/S1056-3911-2010-00569-3

Authors: Dan Abramovich, Martin Olsson and Angelo Vistoli
Journal: J. Algebraic Geom. 20 (2011), 399-477
DOI: https://doi.org/10.1090/S1056-3911-2010-00569-3
Published electronically: September 13, 2010
Erratum: J. Algebraic Geom. 24 (2015), 399-400.
MathSciNet review: 2786662
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Abstract | References | Additional Information

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
MR Author ID: 309312
ORCID: 0000-0003-0719-0989
Email: abrmovic@math.brown.edu

Martin Olsson
Affiliation: Department of Mathematics #3840, University of California, Berkeley, California 94720-3840
MR Author ID: 718643
Email: molsson@math.berkeley.edu

Angelo Vistoli
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
MR Author ID: 194370
ORCID: 0000-0003-3857-3755
Email: angelo.vistoli@sns.it

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