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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 $$\sqrt{J}$$.
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Projectivity of the moduli of equidimensional branchvarieties
We resolve an open problem posed by Alexeev-Knutson on the projectivity of the moduli of branchvarieties in the equidimensional case. As an application, we construct projective moduli spaces of reduced equidimensional varieties equipped with ample linear series and subject to a semistability condition.
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- PAR ID:
- 10676107
- Publisher / Repository:
- Duke University Press
- Date Published:
- Journal Name:
- Duke Mathematical Journal
- Volume:
- 175
- Issue:
- 3
- ISSN:
- 0012-7094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1215/00127094-2025-0032
