The moduli spaces of bielliptic curves of genus 4 with more bielliptic structures

Let $C$ be an irreducible, smooth, projective curve of genus $g\ge3$ over the complex field $\Bbb C$. The curve $C$ is called {\sl bielliptic}\/ if it admits a degree two morphism $\pi\colon C\to E$ onto an elliptic curve $E$: such a morphism is called a {\sl bielliptic structure}\/ on $C$. If $C$ is bielliptic and $g\ge6$ then the bielliptic structure on $C$ is unique, but if $g=3,4,5$ this holds true only generically and there are curves carrying $n>1$ bielliptic structures. We have the sharp bounds $n\le 21,10,5$ for $g=3,4,5$ respectively. Let ${\frak M}_g$ be the coarse moduli space of irreducible, smooth, projective curves of genus $g=3,4,5$. We denote by ${\frak B}_g^n$ the locus of points in ${\frak M}_g$ representing curves carrying at least $n$ bielliptic structures. It is then natural to ask the following questions. Clearly ${\frak B}_g^n\subseteq {\frak B}_g^{n-1}$: does ${\frak B}_g^n\ne {\frak B}_g^{n-1}$ hold? What are the irreducible components of ${\frak B}_g^n$? Are the irreducible components of ${\frak B}_g^n$ rational? How do the irreducible components of ${\frak B}_g^n$ intersect each other? Let $C\in{\frak B}_g^2$: how many non-isomorphic elliptic quotient can it have? In the present paper we give complete answers to the above questions in the case $g=4$.