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

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

Article copyright:
© Copyright 2002
American Mathematical Society