Let $M\_{g,[n]}$ be the moduli space of curves over complex number with level-$n$ structure. It is known that when $n\geq 3$, there is a universal family of curves over it: $f:\,C\_{g,[n]} \to M\_{g,[n]}$. There is a natural involution $\tau$ on $M\_{g,[n]}$ defined by $\tau\big([C, \alpha]\big) = [C,-\alpha]$, where $C$ is a curve, $\alpha$ is a level structure, and $[C,\alpha]$ is the corresponding point in $M\_{g,[n]}$. The fixed points are exactly corresponding to the hyperelliptic locus $H\_{g,[n]}$. (See Oort, F.; Steenbrink, J.;
{The local Torelli problem for algebraic curves},
Journ\'ees de G\'eometrie Alg\'ebrique d'Angers, (1979), 157--204.)


My question is that, Is there an involution $\sigma$ on $C\_{g,[n]}$ such that: (i) $f\circ \sigma=\tau\circ f$; (ii) $\tau|\_{C\_p}$ is the hyperelliptic involution of $C\_p$, where $p\in H\_{g,[n]}$ and $C\_p \subseteq C\_{g,[n]}$ the fiber over $p$.



Involutions on Moduli space of curves $C_{g,[n]} \to M_{g,[n]}$
Yes. $C\_{g,[n]}$ is a fine moduli space for curves with one marked point + level $n$ structure. Just define $\sigma $ by $\sigma ([C,p,\alpha ])=[C,p,-\alpha ]$. If $C$ is hyperelliptic, the hyperelliptic involution $\iota $ induces an isomorphism $[C,p,-\alpha ]\cong [C,\iota (p), \alpha ]$, so $\sigma \_{|C}=\iota $.


