Jacobian schemes arising from hypersurface arrangements in $\mathbb P^n$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913426003984384 |
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| author | Migliore, Juan Nagel, Uwe |
| author_facet | Migliore, Juan Nagel, Uwe |
| contents | Freeness is an important property of a hypersurface arrangement, although its presence is not well understood. A hypersurface arrangement in $\PP^n$ is free if $S/J$ is Cohen-Macaulay (CM), where $S = K[x_0,\ldots,x_n]$ and $J$ is the Jacobian ideal. We study three related unmixed ideals: $J^{top}$, the intersection of height two primary components, $\sqrt{J^{top}}$, the radical of $J^{top}$, and when the $f_i$ are smooth we also study $\sqrt{J}$. Under mild hypotheses, we show that these ideals are CM. This establishes a full generalization of an earlier result with Schenck from hyperplane arrangements to hypersurface arrangements. If the hypotheses fail for an arrangement in projective $3$-space, the Hartshorne-Rao module measures the failure of CMness. We establish consequences for the even liaison classes of $J^{top}$ and $\sqrt{J}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01192 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Jacobian schemes arising from hypersurface arrangements in $\mathbb P^n$ Migliore, Juan Nagel, Uwe Algebraic Geometry Commutative Algebra 14N20, 14M05, 14M06, 32S22, 13C40, 13H10 Freeness is an important property of a hypersurface arrangement, although its presence is not well understood. A hypersurface arrangement in $\PP^n$ is free if $S/J$ is Cohen-Macaulay (CM), where $S = K[x_0,\ldots,x_n]$ and $J$ is the Jacobian ideal. We study three related unmixed ideals: $J^{top}$, the intersection of height two primary components, $\sqrt{J^{top}}$, the radical of $J^{top}$, and when the $f_i$ are smooth we also study $\sqrt{J}$. Under mild hypotheses, we show that these ideals are CM. This establishes a full generalization of an earlier result with Schenck from hyperplane arrangements to hypersurface arrangements. If the hypotheses fail for an arrangement in projective $3$-space, the Hartshorne-Rao module measures the failure of CMness. We establish consequences for the even liaison classes of $J^{top}$ and $\sqrt{J}$. |
| title | Jacobian schemes arising from hypersurface arrangements in $\mathbb P^n$ |
| topic | Algebraic Geometry Commutative Algebra 14N20, 14M05, 14M06, 32S22, 13C40, 13H10 |
| url | https://arxiv.org/abs/2312.01192 |