On the Bielliptic and Bihyperelliptic Loci

Abstract

We study some particular loci inside the moduli space ${\mathcal{M}}_g$, namely the bielliptic locus (i.e. the locus of curves admitting a $2:1$ cover over an elliptic curve $E$) and the bihyperelliptic locus (i.e. the locus of curves admitting a $2:1$ cover over a hyperelliptic curve $C\text{'}$, $g\left(C\text{'}\right)\ge 2$). We show that the bielliptic locus is not a totally geodesic subvariety of ${\mathcal{A}}_g$ if $g\ge 4$ (whereas it is for $g=3$, see [18]) and that the bihyperelliptic locus is not totally geodesic in ${\mathcal{A}}_g$ if $g\ge 3g\text{'}$. We also give a lower bound for the rank of the second Gaussian map at the generic point of the bielliptic locus and an upper bound for this rank for every bielliptic curve.