The arithmetic rank of the residual intersections of a complete intersection ideal
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915941262032896 |
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| author | Batavia, Manav Sundaram, Kesavan Mohana Pandey, Vaibhav Murray, Taylor |
| author_facet | Batavia, Manav Sundaram, Kesavan Mohana Pandey, Vaibhav Murray, Taylor |
| contents | The arithmetic rank of an ideal in a polynomial ring over an algebraically closed field is the smallest number of equations needed to define its vanishing locus set-theoretically. We determine the arithmetic rank of the generic $m$-residual intersection of an ideal generated by $n$ indeterminates for all $m\geq n$ and in every characteristic. We further give an explicit description of its set-theoretic generators. Our main result provides a sharp upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in any Noetherian local ring. In particular, given a complete intersection ideal of height at least two, any of its generic residual intersections -- including its generic link -- fails to be a set-theoretic complete intersection in characteristic zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17049 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The arithmetic rank of the residual intersections of a complete intersection ideal Batavia, Manav Sundaram, Kesavan Mohana Pandey, Vaibhav Murray, Taylor Commutative Algebra Algebraic Geometry Primary 13C40, Secondary 13A35, 13A50, 13D45, 14F20 The arithmetic rank of an ideal in a polynomial ring over an algebraically closed field is the smallest number of equations needed to define its vanishing locus set-theoretically. We determine the arithmetic rank of the generic $m$-residual intersection of an ideal generated by $n$ indeterminates for all $m\geq n$ and in every characteristic. We further give an explicit description of its set-theoretic generators. Our main result provides a sharp upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in any Noetherian local ring. In particular, given a complete intersection ideal of height at least two, any of its generic residual intersections -- including its generic link -- fails to be a set-theoretic complete intersection in characteristic zero. |
| title | The arithmetic rank of the residual intersections of a complete intersection ideal |
| topic | Commutative Algebra Algebraic Geometry Primary 13C40, Secondary 13A35, 13A50, 13D45, 14F20 |
| url | https://arxiv.org/abs/2510.17049 |