Schemes Supported on the Singular Locus of a Hyperplane Arrangement in Pn

Juan Migliore, Uwe Nagel, Henry Schenck

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A hyperplane arrangement in Pn is free if R/J is Cohen-Macaulay (CM), where R = k[x0, . . . , xn] and J is the Jacobian ideal. We study the CM-ness of two related unmixed ideals: Jun, the intersection of height two primary components, and √ J, the radical. Under a mild hypothesis, we show these ideals are CM. Suppose the hypothesis fails. For equidimensional curves in P3, the Hartshorne-Rao module measures the failure of CMness and determines the even liaison class of the curve. We show that for any positive integer r, there is an arrangement for which R/Jun (resp. R/ √ J) fails to be CM in only one degree, and this failure is by r. We draw consequences for the even liaison class of Jun or √ J.

Original languageEnglish
Pages (from-to)140-170
Number of pages31
JournalInternational Mathematics Research Notices
Issue number1
StatePublished - Jan 1 2022

Bibliographical note

Funding Information:
This work was supported by the Simons Foundation [309556 to J.M., 317096 and 636513 U.N.]; and

Funding Information:
the National Science Foundation [1818646 to H.S.].

Publisher Copyright:
© The Author(s) 2020.

ASJC Scopus subject areas

  • Mathematics (all)


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