The rationality of the moduli spaces of bielliptic curves of genus five with more bielliptic structures (Q1880818)

A bielliptic curve \(C\) is a curve admitting a degree two morphism \(\pi\): \(C\to E\) onto an elliptic curve. Such a morphism \(\pi\) is called a bielliptic structure on \(C\). If the genus of \(C\) is five, then \(C\) may admit \(1,2,3\) or 5 distinct bielliptic structures. Let \({\mathcal M}^{\text{be},n}_5\) be the locus, inside the coarse moduli space of a smooth curve \({\mathcal M}_5\), representing the isomorphism classes of those curves having at least \(n\) bielliptic structures. It is known that \({\mathcal M}^{\text{be},n}_5\) is a rational variety. The current paper shows that \({\mathcal M}^{\text{be},n}_5\) is rational also for the other values of \(n\).