Zero-cycles with modulus associated to hyperplane arrangements on affine spaces

Abstract

For a hyperplane arrangement ${\mathcal {A}}$ on the affine n-space over k, we define and study a group of zero-cycles relative to ${\mathcal {A}}$, which is closely related to the relative Chow group of M. Kerz and S. Saito. We compute our cycle groups for a special kind of hyperplane arrangements, called polysimplicial spheres. We prove that they are isomorphic to the Milnor K-groups $K_n ^M (k)$, similar to the theorem of Nesterenko–Suslin–Totaro. Using this result, we show that the Kerz–Saito relative Chow group $\mathrm{CH}_0 ({\mathbb {A}}^n|D)$ does not necessarily vanish for $\deg (D_\mathrm{red}) > n \ge 2$, contrary to the result of Krishna–Park that for $n \ge 2$ and $\deg (D_{\mathrm{red}}) \le n$, the group does vanish when k is a field of characteristic 0.