On the non-divisorial base locus of big and nef line bundles of K3^{[2]}-type varieties

TL;DR

This paper investigates the base loci of big and nef line bundles on irreducible symplectic varieties, revealing new behaviors in higher dimensions and showing that generic polarizations are base point free.

Contribution

It determines the base loci of all big and nef line bundles on certain K3^{[2]}-type varieties, including examples with non-trivial codimension two base loci.

Findings

Existence of an ample line bundle with a non-trivial base locus in codimension two.

Generic polarizations in moduli spaces are base point free.

Complete characterization of base loci on Hilbert schemes of two points on K3 surfaces.

Abstract

We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized K3$^{[2]}$-type varieties, the polarization is base point free.