On the rational cohomology of moduli spaces of Prym curves

We investigate the low degree rational cohomology groups of the moduli space of twisted Prym curves ${\overline{Pr}_{g,n}^{\hspace{0.05cm}(m_1, \ldots, m_n)}} $, where the integer twists $0\leq m_i\leq 1$ have even sum over $i$.

We prove that these groups vanish in odd degree $\leq3$ and that the group in degree $2$ is algebraic.

In particular, the results cover the classical moduli spaces of Prym curves and Prym curves with simple ramifications.