Irreducible unirational and uniruled components of moduli spaces of polarized Enriques surfaces

We give an explicit description of the irreducible components of the modulispaces of polarized Enriques surfaces in terms of decompositions of thepolarization as an effective sum of isotropic classes. We prove that infinitelymany of these components are unirational (resp. uniruled). In particular, thisapplies to components of arbitrarily large genus $g$ and $phi$-invariant ofthe polarization.