On the Picard group of the moduli scheme of stable curves in positive characteristic
For every prime \(p\) let denote by \(M_g^-(p)\) the moduli scheme of stable curves of genus \(g\) over an algebraically closed field of characteristic \(p\). In this paper it is proved the following theorem: Fix an integer \(g\geq 5\) and a prime \(p> 84(g-1)\), then the Néron-Severi group \(\text{NS} (M_g^-(p))\) of \(M_g^-(p)\) is freely generated by the \([g/2]+2\) classes \(\lambda\) and \(\Delta_i\), \(0\leq i\leq [g/2]\). The proof uses a reduction to the characteristic 0 case (i.e. to the Harer's topological theorem) and a result by \textit{M. Pikaart} and \textit{A. J. de Jong} [The moduli space of curves, Proc. Conf. Texel Island 1994, Prog. Math. 129, 483-509 (1995; Zbl 0860.14024)].
scientific article; zbMATH DE number 1121643
scientific article