Monodromy of the Prym map and semicanonical pencils in genus 6

The Prym map P₆ in genus 6 is dominant and generically finite of degree 27. When restricted to the divisor of curves with an odd semicanonical pencil T₆ᵒ, it is still generically finite, but of degree strictly smaller. In this paper, we prove that P₆ restricted to T₆ᵒ is birational and that the monodromy group over the image of T₆ᵒ is the Weyl group WD₅. Thus, there are two other irreducible divisors in the moduli space of Prym curves R₆ and the degree of P₆ restricted to them is 10 and 16. Moreover, we study the geometry of the divisor where P₆ has degree 10.