Moduli of \(G\)-covers of curves: geometry and singularities. (Modules de \(G\)-recouvrements: géométrie et singularités.) (English. French summary) Zbl 1512.14017
In this paper the author studies the birational geometry of the moduli space of curves with a \(G\)-cover, in particular where \(G=S_3\). For background on the compactification, \(\overline{\ M}_g\) of the moduli space of smooth curves of genus \(g\) see [P. Deligne and D. Mumford, Publ. Math., Inst. Hautes Étud. Sci. 36, 75–109 (1969; Zbl 0181.48803)]. Eisenbud, Harris and Mumford proved that \(\overline{\mathcal{M}}_g\) is a variety of general type for genus \(g>23\) [D. Eisenbud and J. Harris, Invent. Math. 90, 359–387 (1987; Zbl 0631.14023)]. Recently it was also shown that \(\overline{\mathcal{M}}_{22}\) and \(\overline{\mathcal{M}}_{23}\) are also of general type, see [G. Farkas et al., “The Kodaira dimensions of \(\overline{\mathcal{M}}_{22}\) and \(\overline{\mathcal{M}}_{23}\)”, Preprint, arXiv:2005.00622].
The author of the present paper generalizes an approach of A. Chiodo and G. Farkas [J. Eur. Math. Soc. (JEMS) 19, No. 3, 603–658 (2017; Zbl 1398.14032)]. He introduces two notions of covers: the twisted G-covers and the admissible G-covers. These two notions are equivalent as shown in [D. Abramovich et al., Commun. Algebra 31, No. 8, 3547–3618 (2003; Zbl 1077.14034)] after giving all the necessary background he proceeds by defining the T-curves and the J-curves and shows that for \(G=S_3\) the J-locus is empty. In a forthcoming second paper the author will show that the moduli space of genus \(g\) connected twisted \(S_3\)-covers is of general type for any odd genus \(g\geq 13\).
The author of the present paper generalizes an approach of A. Chiodo and G. Farkas [J. Eur. Math. Soc. (JEMS) 19, No. 3, 603–658 (2017; Zbl 1398.14032)]. He introduces two notions of covers: the twisted G-covers and the admissible G-covers. These two notions are equivalent as shown in [D. Abramovich et al., Commun. Algebra 31, No. 8, 3547–3618 (2003; Zbl 1077.14034)] after giving all the necessary background he proceeds by defining the T-curves and the J-curves and shows that for \(G=S_3\) the J-locus is empty. In a forthcoming second paper the author will show that the moduli space of genus \(g\) connected twisted \(S_3\)-covers is of general type for any odd genus \(g\geq 13\).
Reviewer: Aigli Papantonopoulou (Ewing)
MSC:
| 14H10 | Families, moduli of curves (algebraic) |
Keywords:
moduli of curvesSoftware:
KoszulDivisorOnPic14M8References:
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