Towards generic base-point-freeness for hyperk\"ahler manifolds of generalized Kummer type

We study base-point-freeness for big and nef line bundles on hyperk\"ahler manifolds of generalized Kummer type: For $n\in \{2,3,4\}$, we show that, generically in all but a finite number of irreducible components of the moduli space of polarized $\mathrm{Kum}^n$-type varieties, the polarization is base-point-free. We also prove generic base-point-freeness in the moduli space in all dimensions if the polarization has divisibility one.