On Fujita's conjecture for a general hyperkähler manifold in the standard series of examples

For the moduli spaces ${\mathrm{\Sigma }}_{d,t}^n$ and ${\mathrm{{\rm Y}}}_{d,t}^n$ of polarized hyperkähler manifolds of Hilb${}^n$(K3)-type and Kum${}^n$-type respectively, with polarization with square $2d$ and divisibility $t$, we study general base point freeness and very ampleness of the polarization. We provide cases where these moduli space are connected and a formula characterizing when these spaces are non-empty.