No Positive Margin from Pencils: Xiao's Lemma, a Rank-r Margin Identity, and the Nef Barrier in Parshin-Moret-Bailly Height Arguments

Parshin–Moret-Bailly height arguments need a positive geometric margin from a pencil of sections; this note shows that the margin is never positive, proves Xiao's lemma for pencils as an exact identity, and applies it to a recent preprint on the abc conjecture. Parshin–Moret-Bailly (PMB) arguments seek an upper bound for the arithmetic self-intersection of the relative dualizing sheaf on the fibers of a fibration over a curve. A recent implementation (C. Soulé, arXiv:2611.05217, version 1) obtains its key estimate from a pencil: two sections of an auxiliary line bundle without common zero, Minkowski's theorem on successive minima, and a transfer from classical to arithmetic degrees. The leading coefficient of the resulting inequality is the geometric margin M = (−L² + m·deg φ*L)/ω², and an upper bound requires M > 0. This note is a companion paper to the abc landscape (Paper A, concept DOI 10.5281/zenodo.21916900); the landscape records the route, this note states the result in citable form. Main results. In rank two, the Koszul resolution of the evaluation map gives the exact identity −L² + m·deg φ*L = −ℓ(Z) − 2m·deg V ≤ 0 (Xiao's lemma for pencils). In rank r, a margin identity in terms of the Bogomolov discriminant of the syzygy kernel and a nef lemma in the spirit of Xiao give Mr ≤ 0 whenever φ*L is semistable, and strict or partial negativity in the Harder–Narasimhan-unstable case. As an application, Proposition 9 vi) of arXiv:2611.05217v1 is false: in the Kodaira fibration used there one has k = 4, at most two of the four square roots over each exceptional point are effective, and deg φ*L = −7k + 4keff ≤ −20 < −3k. A second route shows that the generic linear system is base-point-free for every choice of the component and of the square root, so the rank-two identity applies to every bundle used in the pencil step. (i) External preprint input: C. Soulé, "On the abc conjecture", arXiv:2611.05217, version 1 (all references concern version 1; a later version may differ); (ii) Structural component: rank-two identity, rank-r margin identity, nef lemma, Harder–Narasimhan dichotomy, all proved under explicitly stated hypotheses; (iii) Review status: adversarial referee review by a different AI provider than the drafting model, a second independent review of the base-point-freeness theorem, and a claim register with automated linting; see the AI disclosure in the paper; (iv) Open question: the arithmetic side of the Harder–Narasimhan-unstable case for rank r ≥ 3 is stated as an open problem. Claim boundary: this note makes no statement about the abc conjecture, about Parshin's conjecture, or about the truth of Theorems 1, A or B of arXiv:2611.05217 as statements. It shows that the pencil step of a PMB argument cannot produce a positive margin under the stated hypotheses, and that the proof of Theorem 1 given in version 1 of that preprint is incomplete at Proposition 9 vi). The author of the preprint was informed of this result before publication. This version is a research draft; revisions are not excluded. Changes in Version 0.1 (November 2026) Initial release. Rank-two identity (Xiao's lemma for pencils) and rank-r margin identity with nef lemma and Harder–Narasimhan dichotomy. Application to arXiv:2611.05217v1: degree formula, k = 4, at most two effective square roots per exceptional point, Proposition 9 vi) false; unconditional base-point-freeness of the generic linear system. Remarks on cross terms, single sections and jets along curves, and source bundles not pulled back from the base; open problem for the unstable case in rank r ≥ 3. Companion paper to the abc landscape (Paper A, 10.5281/zenodo.21916900); sources on GitHub: research-line/abc-hct. DE/EN: English and German PDFs published synchronously (German title: "Kein positiver Spielraum aus Büschelargumenten: Xiaos Lemma, eine Spielraumidentität in Rang r und die Nef-Barriere in Höhenargumenten nach Parshin und Moret-Bailly").