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Moduli of twisted spin curves


Authors: Dan Abramovich and Tyler J. Jarvis
Journal: Proc. Amer. Math. Soc. 131 (2003), 685-699
MSC (2000): Primary 14H10
DOI: https://doi.org/10.1090/S0002-9939-02-06562-0
Published electronically: July 17, 2002
MathSciNet review: 1937405
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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we give a new, natural construction of a compactification of the stack of smooth $r$-spin curves, which we call the stack of stable twisted $r$-spin curves. This stack is identified with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible $\mathbb G_{\mathbf{m}}$-spaces and $\mathbb Q$-line bundles are given as well. The infinitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves.

We construct representable morphisms from the stacks of stable twisted $r$-spin curves to the stacks of stable $r$-spin curves and show that they are isomorphisms. Many delicate features of $r$-spin curves, including torsion free sheaves with power maps, arise as simple by-products of twisted spin curves. Various constructions, such as the $\bar\partial$-operator of Seeley and Singer and Witten's cohomology class go through without complications in the setting of twisted spin curves.


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Additional Information

Dan Abramovich
Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
Email: abrmovic@math.bu.edu

Tyler J. Jarvis
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: jarvis@math.byu.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06562-0
Received by editor(s): April 13, 2001
Received by editor(s) in revised form: October 11, 2001
Published electronically: July 17, 2002
Additional Notes: The first author’s research was partially supported by NSF grants DMS-9700520 and DMS-0070970
The second author’s research was partially supported by NSA grant MDA904-99-1-0039
Communicated by: Michael Stillman
Article copyright: © Copyright 2002 American Mathematical Society