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: 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 by-products 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.

**1.**D. Abramovich and A. Vistoli,*Compactifying the Space of Stable Maps*, J. Amer. Math. Soc.**15**(2002), 27-75.**2.**D. Abramovich, A. Corti and A. Vistoli,*Twisted bundles and admissible covers*, preprint (2001),`math.AG/0106211`.**3.**P. Deligne and D. Mumford,*The Irreducibility of the Space of Curves of Given Genus,*Inst. Hautes Études Sci. Publ. Math. No.**36**(1969), 75-109. MR**41:6850****4.**T. Jarvis,*Torsion-Free Sheaves and Moduli of Generalized Spin Curves*, Compositio Math. 110 (1998), no. 3, 291-333. MR**99b:14026****5.**T. Jarvis,*Geometry of the moduli of higher spin curves,*Internat. J. of Math.**11**(2000), no. 5, 637-663. MR**2001f:14050****6.**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), 167-177, Contemp. Math.,**276**, Amer. Math. Soc., Providence, RI, 2001.**7.**T. Jarvis, T. Kimura and A. Vaintrob,*Moduli spaces of higher spin curves and integrable hierarchies*. Compositio Math.,**126**(2001), no. 2, 157-212.**8.**T. Mochizuki,*The Virtual Class of the Moduli Stack of -spin Curves*, preprint, (2001).`http://math01.sci.osaka-cu.ac.jp/takuro/list.html`**9.**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), 229-249, Contemp. Math.,**276**, Amer. Math. Soc., Providence, RI, 2001.**10.**R. Seeley and I. M. Singer,*Extending to Singular Riemann Surfaces*, J. Geom. Phys 5 (1988), no.1, 121-136. MR**91f:58101****11.**E. Witten,*Algebraic Geometry Associated with Matrix Models of Two-dimensional Gravity*, Topological models in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX 1993, 235-269. MR**94c:32012**

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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