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Moduli of twisted spin curves. (English) Zbl 1037.14008
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.

MSC:
14H10 Families, moduli of curves (algebraic)
14D20 Algebraic moduli problems, moduli of vector bundles
14A20 Generalizations (algebraic spaces, stacks)
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