A seven-hyperplane counterexample to a generic-initial degree bound

For the characteristic-zero arrangement Q=xy(x-y)(x-2y)zw(x+z+w), the reverse-lexicographic generic initial ideal of its ambient Jacobian ideal has pure-second-variable cutoff 8 and a degree 7 minimal generator whose largest variable is the third. A self-contained open-set proof refutes the general-dimensional degree bound of Bigatti–Palezzato–Torielli, also stated in OWR Report 5/2021. The valid three-variable arrangement theorem remains intact. The construction is a classical parallel connection; logarithmic, Hilbert–Burch and generic-section methods are prior, and the applicable Migliore–Nagel resolution of the distinct unmixed ideal is credited. A bounded audit through 15 October 2026 found no complete predecessor in its inspected corpus; historical priority and independent discovery remain unestablished. Ilardi (2018) full text was not obtained, and parameter families and historical code were not exhausted. The deposit includes the six-page note, LaTeX source and exact standard-library Python verification certificates and authored audits. AI tools were used extensively. This preprint is unrefereed and has not undergone conventional human peer review.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-15
DOI
https://doi.org/10.5281/zenodo.23395702
Primary Topic
Commutative Algebra and Its Applications
Type
preprint
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preprint

A seven-hyperplane counterexample to a generic-initial degree bound

Kriebel Alec
Zenodo (CERN European Organization for Nuclear Research)
Commutative Algebra and Its Applications
preprint

A seven-hyperplane counterexample to a generic-initial degree bound

Kriebel Alec
preprint en

Abstract

For the characteristic-zero arrangement Q=xy(x-y)(x-2y)zw(x+z+w), the reverse-lexicographic generic initial ideal of its ambient Jacobian ideal has pure-second-variable cutoff 8 and a degree 7 minimal generator whose largest variable is the third. A self-contained open-set proof refutes the general-dimensional degree bound of Bigatti–Palezzato–Torielli, also stated in OWR Report 5/2021. The valid three-variable arrangement theorem remains intact. The construction is a classical parallel connection; logarithmic, Hilbert–Burch and generic-section methods are prior, and the applicable Migliore–Nagel resolution of the distinct unmixed ideal is credited. A bounded audit through 15 October 2026 found no complete predecessor in its inspected corpus; historical priority and independent discovery remain unestablished. Ilardi (2018) full text was not obtained, and parameter families and historical code were not exhausted. The deposit includes the six-page note, LaTeX source and exact standard-library Python verification certificates and authored audits. AI tools were used extensively. This preprint is unrefereed and has not undergone conventional human peer review.

Zenodo (CERN European Organization for Nuclear Research)
Commutative Algebra and Its Applications
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