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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.
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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