On the Bielliptic and bihyperelliptic loci

TL;DR

This paper investigates the geometric properties of bielliptic and bihyperelliptic loci within the moduli space of curves, demonstrating they are not totally geodesic in the ambient space for certain genera and providing bounds on the second gaussian map's rank.

Contribution

It establishes non-totally geodesic nature of these loci in the moduli space for specific genera and offers bounds on the second gaussian map's rank for bielliptic curves.

Findings

Bielliptic locus is not totally geodesic for g ≥ 4.

Bihyperelliptic locus is not totally geodesic if g ≥ 3g'.

Provides bounds on the second gaussian map's rank for bielliptic curves.

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 $C'$, $g(C') \geq 2$). We show that the bielliptic locus is not a totally geodesic subvariety of $\mathcal{A}_g$ if $g \geq 4$ (while it is for $g=3$, see [16]) and that the bihyperelliptic locus is not totally geodesic in $\mathcal{A}_g$ if $g \geq 3g'$. 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.