On the rationality problem for moduli space of spin hyperelliptic curves with one marked point and of other special moduli of curves

In this talk we show how the rationality problem for the moduli space of spin hyperelliptic curve or for the moduli space of genus-$4$ spin curves with a theta-vanishing or for others special moduli space of spin curves can be studied using the technique built with Hiromichi Takagi to show the rationality of spin curves of genus $4$.