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.

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关于 K3[2] 型品种 big 和 nef 系丛的非除数基轨迹

我们在不可约辛簇上逼近 big 和 nef 系丛的非除数基础基因座。虽然对于 K3 表面，只能出现除数碱基基因座，但对于更一般的不可约辛变体的非除数碱基基因座的行为一无所知。我们确定所有 big 和 nef 线束的基本轨迹，基于属二的非常一般的 K3 曲面上的两点的希尔伯特方案及其双有理模型。值得注意的是，我们在第二维中发现了一个具有非平凡基本轨迹的充足线束。我们推断，一般在极化 K3 的模空间中[2]型品种，极化无基点。