On Fujita's conjecture for a general hyperk\"ahler manifold in the standard series of examples

Abstract

For the moduli spaces $\Sigma_{d,t}^n$ and $\Upsilon_{d,t}^n$ of polarized hyperk\"ahler 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.