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

Juan Migliore; Uwe Nagel; Henry Schenck

doi:10.1093/imrn/rnaa113

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

Juan Migliore, Uwe Nagel, Henry Schenck

Mathematics

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1 Scopus citations

Abstract

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 language	English
Pages (from-to)	140-170
Number of pages	31
Journal	International Mathematics Research Notices
Volume	2022
Issue number	1
DOIs	https://doi.org/10.1093/imrn/rnaa113
State	Published - 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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10.1093/imrn/rnaa113

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abstract = "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.",

author = "Juan Migliore and Uwe Nagel and Henry Schenck",

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N2 - 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.

AB - 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.

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

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