Rationality of \(\mathcal{R}_2\) and \(\mathcal{R}_3\)
Let \(R_g\) be the moduli space of pairs \((C,\epsilon)\), where \(C\) is a curve of genus \(g\) and \(\epsilon\) is a non-zero torsion-two divisor on \(C\). The goal of this paper is to present the original author's proofs for the rationality of \(R_2\) and \(R_3\). As the author writes in the Introduction, the proof of the rationality of \(R_2\) given here is perhaps the first published proof of this fact. The first published proof of the rationality of \(R_3\) is given in 1992 by \textit{P. Katsylo} [Aspects Math. E 25, 61--65 (1994; Zbl 0829.14015)]. The proof of the rationality of \(R_3\) in the present paper follows an unpublished manuscript of the author since 1990, where is given an alternative proof of the rationality of \(R_3\) based on a suggestion of \textit{A. Del Centina} and \textit{S. Recillas} [Lect. Notes Math. 1414, 28--50 (1989; Zbl 0722.14012)].
scientific article; zbMATH DE number 5296117