## Moduli of twisted spin curves

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- by Dan Abramovich and Tyler J. Jarvis PDF
- Proc. Amer. Math. Soc.
**131**(2003), 685-699 Request permission

## 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.## References

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## Additional Information

**Dan Abramovich**- Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
- MR Author ID: 309312
- ORCID: 0000-0003-0719-0989
- Email: abrmovic@math.bu.edu
**Tyler J. Jarvis**- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- Email: jarvis@math.byu.edu
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 685-699 - MSC (2000): Primary 14H10
- DOI: https://doi.org/10.1090/S0002-9939-02-06562-0
- MathSciNet review: 1937405