Moduli of twisted spin curves
Authors:
Dan Abramovich and Tyler J. Jarvis
Journal:
Proc. Amer. Math. Soc. 131 (2003), 685699
MSC (2000):
Primary 14H10
Published electronically:
July 17, 2002
MathSciNet review:
1937405
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Abstract: In this note we give a new, natural construction of a compactification of the stack of smooth spin curves, which we call the stack of stable twisted 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 spaces and 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 spin curves to the stacks of stable spin curves and show that they are isomorphisms. Many delicate features of spin curves, including torsion free sheaves with power maps, arise as simple byproducts of twisted spin curves. Various constructions, such as the 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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 D. Abramovich and A. Vistoli, Compactifying the Space of Stable Maps, J. Amer. Math. Soc. 15 (2002), 2775.
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 D. Abramovich, A. Corti and A. Vistoli, Twisted bundles and admissible covers, preprint (2001), math.AG/0106211.
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 T. Jarvis, TorsionFree Sheaves and Moduli of Generalized Spin Curves, Compositio Math. 110 (1998), no. 3, 291333. MR 99b:14026
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 T. Jarvis, T. Kimura, and A. Vaintrob, Gravitational Descendants and the Moduli Space of Higher Spin Curves. In E. Previato (Ed.), Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000), 167177, Contemp. Math., 276, Amer. Math. Soc., Providence, RI, 2001.
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 A. Polishchuk and A. Vaintrob Algebraic Construction of Witten's Top Chern Class. In E. Previato (Ed.), Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000), 229249, Contemp. Math., 276, Amer. Math. Soc., Providence, RI, 2001.
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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:
http://dx.doi.org/10.1090/S0002993902065620
PII:
S 00029939(02)065620
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 DMS9700520 and DMS0070970
The second author’s research was partially supported by NSA grant MDA9049910039
Communicated by:
Michael Stillman
Article copyright:
© Copyright 2002
American Mathematical Society
