On a subvariety of the moduli space
The moduli space \({\mathcal M}_g\) of compact Riemann surfaces (complex algebraic curves) of genus \(g\geq 2\) has a singularity locus that consists (if \(g> 2\)) of the surfaces with non-trivial automorphism group. It is interesting to study certain irreducible subvarieties \({\mathcal M}_g(H)\) of this singular locus attached to a group \(H\) of mapping-classes [see e.g. \textit{G. Gonzalez-Diez} and \textit{W. J. Harvey}, Lond. Math. Soc. Lect. Note Ser. 173, 75--93 (1992; Zbl 0763.32014)]. In the paper under review the author gives an explicit description of a non-normal irreducible subvariety of the moduli space of Riemann surfaces of genus 3 characterized by a non-cyclic group action. He finds defining equations of the family of hyperelliptic curves of genus 3 whose full automorphism group is \(C_3\times C_4\). This completes the list of full automorphism groups of hyperelliptic curves given in [\textit{E. Bujalance}, \textit{J. M. Gamboa} and \textit{G. Gromadzki}, Manuscr. Math. 79, No. 3--4, 267--282 (1993; Zbl 0788.30031)]. He also corrects a mistake contained in the above quoted article.
scientific article; zbMATH DE number 2156862