Exceptional points in the elliptic-hyperelliptic locus (Q2477644)

An exceptional point in the moduli space \(\mathcal{M}_{g}\) of compact Riemann surfaces of genus \(g\) is a unique surface class whose full automorphism group acts with triangular signature (a group \(G\) acts on a compact Riemann surface \(X\) with triangular signature if the quotient space \(X/G\) has genus \(0\) and the quotient map \(\pi\colon X\rightarrow X/G\) is branched over exactly three points). A surface \(X\) is said to be symmetric if it admits an anticonformal involution, and it is said to be \(k\)-hyperelliptic if it admits a conformal involution, called the \(k\)-hyperelliptic involution, with quotient a curve of genus \(k\). In the special case where \(k=1\), the quotient curve is an elliptic curve and we call \(X\) an elliptic-hyperelliptic curve. Let \(\mathcal{M}^{k}_{g}\) denote the \(k\)-hyperelliptic locus (the surface classes of \(\mathcal{M}_{g}\) admitting a \(k\)-hyperelliptic involution). In the paper under review, the authors prove a number of interesting results about exceptional points, especially those which are also symmetric, in \(\mathcal{M}_{g}^{1}\). First, the authors consider the problem of determining each group \(G\) up to topological equivalence which can act with a triangular signature on a symmetric surface \(X\) and which contains a \(1\)-hyperelliptic involution for genus \(g> 5\). The purpose for the restriction on the genus is that it guarantees the subgroup generated by the \(1\)-hyperelliptic involution is normal in \(G\) and so simplifies the calculations. To determine such groups, the authors utilize some previous results on the classification of topological equivalence classes of elliptic-hyperelliptic actions [\textit{E. Tyszkowska}, J. Algebra 288, No. 2, 345--363 (2005; Zbl 1078.30037)], together with computational methods from \textit{D. Singerman} [Math. Ann. 210, 17--32 (1974; Zbl 0272.30022)] which provides complete conditions for when \(G\) acts on a symmetric surface. Next the authors show that for infinitely many different \(g\), the number of exceptional points in \(\mathcal{M}_{g}^{1}\) is larger than any preassigned integer \(n\), though for any \(g\), at most four are also symmetric (the authors also show that for infinitely many \(g\), the number of exceptional points is \(0\)). This is in stark contrast to the hyperelliptic locus, \(\mathcal{M}_{g}^{0}\) where the number of exceptional points is always between \(3\) and \(5\), and is precisely \(3\) for all \(g>30\), all of which are symmetric [see \textit{A. Weaver}, Geom. Dedicata 103, 69--87 (2004; Zbl 1047.32012)]. It is also in contrast to the general \(k\)-hyperelliptic locus \(\mathcal{M}_{g}^{k}\) where for sufficiently high \(g\), there are no exceptional points. The proof of this result arises from the construction of arbitrarily many different isomorphism classes of such \(G\)-actions for infinitely many \(g\), and then refers to the previous results to illustrate that at most four are symmetric. The authors finish by determining the full group of conformal and anticonformal automorphisms of symmetric exceptional points in \(\mathcal{M}_{g}^{1}\). If \(X\) is a symmetric surface whose surface class is an exceptional point in \(\mathcal{M}_{g}^{1}\), then the existence of a symmetry on \(X\) implies that the full group of conformal and anticonformal automorphisms is a cyclic \(2\) extension of the full group of conformal automorphisms of \(X\). The possible full groups of comformal automorphisms of such a surface were determined in [\textit{E. Tyszkowska}, loc. cit.] (though with an error which has been corrected in the paper under review). The authors use these facts and explicit computations to determine the full groups of conformal and anticomformal automorphisms.