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.

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Correspondence to Jinhyun Park.

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Crisman, KD., Park, J. Zero-cycles with modulus associated to hyperplane arrangements on affine spaces. manuscripta math. 155, 15–45 (2018). https://doi.org/10.1007/s00229-017-0931-x

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Mathematics Subject Classification

  • Primary 14C25
  • Secondary 19E15