The virtual class of the moduli stack of stable \(r\)-spin curves
type: Item
instance of: scholarly article
title: The virtual class of the moduli stack of stable \(r\)-spin curves
zbMATH Open document ID: 1136.14015
author: Takuro Mochizuki
published in: Communications in Mathematical Physics
publication date: 2006-11-29T00:00:00Z
review text: Let \(n\) be a non-negative integer, \(r\) be a natural number and \({\mathfrak m}=(m_1,\dots,m_n)\) an \(n\)-tuple of non-negative integers. A smooth \(n\)-pointed \(r\)-spin curve of genus \(g\) is a triple \((C,{\mathfrak p},{\mathcal F})\) consisting of a smooth genus \(g\) curve \(C\), with \(n\) marked points \({\mathfrak p}=(p_1,\dots,p_n)\) and a line bundle \({\mathcal F}\) with an isomorphism \({\mathcal F}^{\otimes r}\simeq \omega_C(-\sum m_ip_i)\). The moduli space \({\mathcal M}_{g,n}^{1/r,{\mathfrak m}}\) for such objects has been naïvely introduced by \textit{E.~Witten} [in: Topological methods in modern mathematics. Proc. symp. John Milnor's sixtieth birthday, State University of New York, Stony Brook, USA, June 14-June 21, 1991. Houston, TX: Publish or Perish, Inc. 235--269 (1993; Zbl 0812.14017)], together with the description of a virtual fundamental class on a `reasonable' compactification of \({\mathcal M}_{g,n}^{1/r,{\mathfrak m}}\). Later, \textit{T.~J.~Jarvis} [Compos. Math. 110, No. 3, 291--333 (1998; Zbl 0912.14010); Int. J. Math. 11, No. 5, 637--663 (2000; Zbl 1094.14504)] was able to introduce a suitable notion of stable \(r\)-spin curve and to prove that the moduli functor for stable \(n\)-pointed \(r\)-spin curves of genus \(g\) and type \({\mathfrak m}\) is represented by a smooth Deligne-Mumford stack \(\overline{\mathcal M}_{g,n}^{1/r,{\mathfrak m}}\). Also, together with \textit{T. Kimura} and \textit{A. Vaintrob}, \textit{T.~J.~Jarvis} [Contemp. Math. 276, 167--177 (2001; Zbl 0986.81105); Compos. Math. 126, No.2, 157-212 (2001; Zbl 1015.14028)] formulated a set of axioms to be satisfied by the virtual class of \(\overline{\mathcal M}_{g,n}^{1/r,{\mathfrak m}}\); later an algebro-geometric construction of these virtual classes was given by \textit{A. Polishchuk} and \textit{A. Vaintrob} [Contemp. Math. 276, 229--249 (2001; Zbl 1051.14007)] and \textit{A. Polishchuk} [in: Aspects of Mathematics E 36, 253--264 (2004; Zbl 1105.14010)].…
reviewed by: Domenico Fiorenza
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 5076618
zbMATH Keywords: moduli spaces
zbMATH Keywords: moduli stacks
zbMATH Keywords: \(r\)-spin curves
zbMATH Keywords: virtual fundamental classes
MaRDI profile type: Publication
full work available at URL: s00220-006-1538-3
OpenAlex ID: W2082130163
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DOI: 10.1007/S00220-006-1538-3
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description: scientific article; zbMATH DE number 5076618