Classical arrangement counterexamples to summed rough-M semicontinuity
The classical dual Hesse arrangement and the established constant-term scaling of an affine line arrangement to its pencil give an explicit reduced flat delta-constant multibranch deformation with finite simultaneous normalization and a common transverse ball. The central ordinary-nine point has delta 36 and rough K(K+D)=7; every sufficiently small nonzero complex fiber has twelve ordinary triple points, summed delta 36 and summed rough 12. The written proof checks all singularities, analytic flatness, fiberwise normalization, boundary transversality and arbitrarily small representatives. The original classical Fermat extension has summed rough excess n²−4 for every integer n≥3 by a structural incidence proof; n=2 gives equality. This is an explicitly attributed classical corollary and formulation clarification of the unrestricted literal summed reading of Questions 2–3 in Borodzik's 2009 Oberwolfach contribution, printed 2446–2447. The question itself does not literally write a sum; its following fixed-ball discussion supports that reading. De Jong–van Straten 1998 already uses the exact nine-line/twelve-triple arrangement in the delta-constant picture-deformation framework. Plamenevskaya–Starkston 2023 explicitly scales affine line constants to zero, and Borodzik 2014 p. 578 already uses contraction to an ordinary multiple point and its rough value r−2. Hirzebruch 1983 records the classical all-n arrangements. No new construction or deformation method, first-priority assertion, or present-openness certificate is claimed. The inspected sources did not establish earlier literal publication of the exact rough-M 7-to-12 comparison; that does not establish firstness. The central fiber has nine branches, so the example does not settle the explicitly unibranched parametric Conjecture 3.8 in the 2014 published paper. No marked-section result or interchange of rough/fine or shifted multibranch conventions is asserted. This four-page unrefereed note is accompanied by identical editable LaTeX/PDF, all 28 unchanged original PR909 files, seven independent construction/invariant/computation/source/priority/disposition audit families, and an externally anchored closed-inventory verifier with isolated nonoptimized replay. Exact computations supplement the universal analytic proof; finite sampling and hash consistency do not certify the proof, priority or source interpretation. The original recovery record is 1 verified/5; earlier attempt history is unavailable and no new author proof-search turn is claimed. Historical optimized-assertion, inventory and schema weaknesses are documented and guarded separately without rewriting original bytes. AI tools were used extensively in research, writing, verification and adversarial review. AI-assisted independent checks are not independent human or journal peer review; neither is claimed. Raw third-party source bodies are not distributed.
Authors
- Kriebel Alec (ORCID: https://orcid.org/0009-0001-9320-500X)
Publication Details
- Journal
- Zenodo (CERN European Organization for Nuclear Research)
- Published
- 2026-10-14
- DOI
- https://doi.org/10.5281/zenodo.23384937
- Primary Topic
- Algebraic Geometry and Number Theory
- Type
- preprint