Moduli of twisted spin curves. (English) Zbl 1037.14008

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The main object of this article is the introduction of the concept of twisted stable \(r\)-spin curves and its comparison with related topics. The following notations are used: Let \(r>0\) and \(g,n \geq 0\) be fixed integers. Furthermore, let \({\mathbf m}= (m_1,m_2,\ldots,m_n)\) be an \(n\)-tuple of integers. A smooth \(r\)-spin curve of type \({\mathbf m}\) is the data \((C \to S, s_i, {\mathcal L}, c)\) with \(C \to S\) a smooth \(n\)-pointed curve of genus \(g\), with the sections \(s_i\), \({\mathcal L}\) an invertible sheaf on \(C\), and \(c\) an isomorphism between \({\mathcal L}^{\otimes r}\) and \(\omega_{C/S}(-m_1S_1-m_2S_2-\ldots -m_nS_n)\), where \(S_i\) is the image of \(s_i\). The \(r\)-th power map on \({\mathbb G_m}\) defines a 1-morphism \(\kappa_r\) from the classifying stack \({\mathcal B}{\mathbb G_m}\) to itself. If \({\mathbf m}=(-1,-1,\ldots , -1)\), then we obtain a 1-commutative diagram \[ \begin{tikzcd} & \mathcal B\mathbb G_m \ar[dr,"\kappa_r"] & \\ \mathcal C \ar[ur,"\mathcal L"] \ar[rr,"{\omega_{C/S}(S_1 + S_2 +\dots S_n)}" '] && \mathcal B \mathbb G_m\rlap{\,.} \end{tikzcd} \] Twisted stable \(r\)-spin curves are now, aside from some technical details, defined by the above diagram, where instead of smooth curves \(C \to S\) twisted \(n\)-pointed curves are now allowed. Twisted stable \(r\)-spin curves are proposed as a natural compactification of the moduli stack of smooth \(r\)-spin curves. The authors construct the moduli stack of twisted stable \(r\)-spin curves and identify this moduli stack with the stack of twisted stable maps of Abramovich and Vistoli. Furthermore, they investigate the infinitesimal structure of this stack and compare the twisted spin curves with the quasi spin curves of Jarvis. They conclude with remarks about the \(\bar \partial\) operator of Seeley and Singer and Witten’s class.


14H10 Families, moduli of curves (algebraic)
14D20 Algebraic moduli problems, moduli of vector bundles
14A20 Generalizations (algebraic spaces, stacks)
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[1] D. Abramovich and A. Vistoli, Compactifying the Space of Stable Maps, J. Amer. Math. Soc. 15 (2002), 27-75. · Zbl 0991.14007
[2] D. Abramovich, A. Corti and A. Vistoli, Twisted bundles and admissible covers, preprint (2001), math.AG/0106211. · Zbl 1077.14034
[3] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes √Čtudes Sci. Publ. Math. 36 (1969), 75 – 109. · Zbl 0181.48803
[4] T. J. Jarvis, Torsion-free sheaves and moduli of generalized spin curves, Compositio Math. 110 (1998), no. 3, 291 – 333. · Zbl 0912.14010
[5] Tyler J. Jarvis, Geometry of the moduli of higher spin curves, Internat. J. Math. 11 (2000), no. 5, 637 – 663. · Zbl 1094.14504
[6] T. Jarvis, T. Kimura, and A. Vaintrob, Gravitational Descendants and the Moduli Space of Higher Spin Curves. In E. Previato , Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000), 167-177, Contemp. Math., 276, Amer. Math. Soc., Providence, RI, 2001. · Zbl 0986.81105
[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. · Zbl 1015.14028
[8] T. Mochizuki, The Virtual Class of the Moduli Stack of \(r\)-spin Curves, preprint, (2001). http://math01.sci.osaka-cu.ac.jp/\(\sim\)takuro/list.html
[9] A. Polishchuk and A. Vaintrob Algebraic Construction of Witten’s Top Chern Class. In E. Previato , Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000), 229-249, Contemp. Math., 276, Amer. Math. Soc., Providence, RI, 2001. · Zbl 1051.14007
[10] R. Seeley and I. M. Singer, Extending \overline\partial to singular Riemann surfaces, J. Geom. Phys. 5 (1988), no. 1, 121 – 136. · Zbl 0692.30038
[11] Edward Witten, Algebraic geometry associated with matrix models of two-dimensional gravity, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 235 – 269. · Zbl 0812.14017
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