Monodromy of the Prym map and semicanonical pencils in genus 6

The Prym map ${\mathcal{P}}_6$ in genus 6 is dominant and generically finite of degree 27. When restricted to the divisor of curves with an odd semicanonical pencil ${\mathcal{T}}_6^o$, it is still generically finite, but of degree strictly smaller. In this paper, we prove that ${\mathcal{P}}_6$ restricted to ${\mathcal{T}}_6^o$ is birational and that the monodromy group over the image of ${\mathcal{T}}_6^o$ is the Weyl group $W{D}_5$. Thus, there are two other irreducible divisors in the moduli space of Prym curves ${\mathcal{R}}_6$ and the degree of ${\mathcal{P}}_6$ restricted to them is 10 and 16. Moreover, we study the geometry of the divisor where ${\mathcal{P}}_6$ has degree 10.