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In the moduli space $\mathcal{R}_g$ of double \'etale covers of curves of afixed genus $g$, the locus of covers of curves with a semicanonical pencildecomposes as the union of two divisors $\mathcal{T}^e_g$ and$\mathcal{T}^o_g$. Adapting arguments of Teixidor for the divisor of curveshaving a semicanonical pencil, we prove that both divisors are irreducible andcompute their divisor classes in the Deligne-Mumford compactification$\overline{\mathcal{R}}_g$.

The divisors of Prym semicanonical pencils