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- Title
Schemes Supported on the Singular Locus of a Hyperplane Arrangement in ℙn.
- Authors
Migliore, Juan; Nagel, Uwe; Schenck, Henry
- Abstract
A hyperplane arrangement in |$\mathbb P^n$| is free if |$R/J$| is Cohen–Macaulay (CM), where |$R = k[x_0,\dots ,x_n]$| and |$J$| is the Jacobian ideal. We study the CM-ness of two related unmixed ideals: |$ J^{un}$| , the intersection of height two primary components, and |$\sqrt{J}$| , the radical. Under a mild hypothesis, we show these ideals are CM. Suppose the hypothesis fails. For equidimensional curves in |$\mathbb P^3$| , the Hartshorne–Rao module measures the failure of CM-ness and determines the even liaison class of the curve. We show that for any positive integer |$r$| , there is an arrangement for which |$R/J^{un}$| (resp. |$R/\sqrt{J}$|) fails to be CM in only one degree, and this failure is by |$r$|. We draw consequences for the even liaison class of |$J^{un}$| or |$\sqrt{J}$|.
- Subjects
LOCUS (Mathematics); HYPERPLANES; INTEGERS
- Publication
IMRN: International Mathematics Research Notices, 2022, Vol 2022, Issue 1, p140
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnaa113