We study some particular loci inside the moduli space , namely the bielliptic locus (i.e. the locus of curves admitting a cover over an elliptic curve ) and the bihyperelliptic locus (i.e. the locus of curves admitting a cover over a hyperelliptic curve , ). We show that the bielliptic locus is not a totally geodesic subvariety of if (whereas it is for , see ) and that the bihyperelliptic locus is not totally geodesic in if . 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.
"On the Bielliptic and Bihyperelliptic Loci." Michigan Math. J. 69 (3) 571 - 600, August 2020. https://doi.org/10.1307/mmj/1596700820