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We study base-point-freeness for big and nef line bundles on hyperk\"ahlermanifolds 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 modulispace of polarized $\mathrm{Kum}^n$-type varieties, the polarization isbase-point-free. We also prove generic base-point-freeness in the moduli spacein all dimensions if the polarization has divisibility one.

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