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

TL;DR

This paper investigates base-point-freeness of line bundles on hyperk"ahler manifolds of generalized Kummer type, showing generic base-point-freeness in various dimensions and conditions, advancing understanding of their geometric properties.

Contribution

It establishes generic base-point-freeness for big and nef line bundles on hyperk"ahler manifolds of generalized Kummer type across multiple dimensions and divisibility conditions.

Findings

Base-point-freeness holds generically in all but finitely many moduli components for n=2,3,4.

Proves generic base-point-freeness in all dimensions for polarizations with divisibility one.

Results improve understanding of line bundle properties on hyperk"ahler manifolds of Kummer type.

Abstract

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.