\(\overline{\mathcal R}_{15}\) is of general type (Q346717)

A Prym curve is a smooth curve with a \(2\)-torsion point in its Jacobian. Let \(\overline{\mathcal R}_g\) be the compactified moduli space of genus \(g\) Prym curves in the sense of \textit{E. Ballico} et al. [Doc. Math., J. DMV 9, 265--281 (2004; Zbl 1072.14029)] and \textit{M. Cornalba} [in: Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co.. 560--589 (1989; Zbl 0800.14011)]. In [\textit{G. Farkas} and \textit{K. Ludwig}, J. Eur. Math. Soc. (JEMS) 12, No. 3, 755--795 (2010; Zbl 1193.14043)] it was shown that \(\overline{\mathcal R}_g\) is of general type for \(g\geq 14\) and \(g\neq 15\). In this paper the author extends the above range to include the case \(g=15\). The idea is the standard one -- writing the canonical class of \(\overline{\mathcal R}_g\) as a positive linear combination of the Hodge class with some other effective divisor classes, which then implies that the canonical class is big. In order to do this, the author constructs explicitly a geometric divisor on \(\overline{\mathcal R}_g\) and analyzes its divisor class. The construction uses Brill-Noether geometry in the setting of Prym curves.