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 $C^{\prime }$, $g(C^{\prime })\ge 2$). We show that the bielliptic locus is not a totally geodesic subvariety of ${\mathcal{A}}_g$ if $g\ge 4$ (while it is for $g=3$, see [16]) and that the bihyperelliptic locus is not totally geodesic in ${\mathcal{A}}_g$ if $g\ge 3g^{\prime }$. We also give a lower bound for the rank of the second gaussian map on the generic point of the bielliptic locus and an upper bound for this rank for every bielliptic curve.

Final version. To appear in the Michigan Mathematical Journal