Exceptional points in the elliptic-hyperelliptic locus

An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is called elliptic-hyperelliptic; one admitting an anticonformal involution is called symmetric. In this paper, we determine, up to topological conjugacy, the full group of conformal and anticonformal automorphisms of a symmetric exceptional point in the elliptic-hyperelliptic locus. We determine the number of ovals of any symmetry of such a surface. We show that while the elliptic-hyperelliptic locus can contain an arbitrarily large number of exceptional points, no more than four are symmetric.