--- title: "A sharp four-factor Cohen–Macaulay obstruction to integral Schubert deformation" date: 2026-09-04 version: "0.1.0-candidate" doi: 10.5281/zenodo.22304827 pdf: https://github.com/ipitchford/sharp-four-factor-cohen-macaulay-obstruction/releases/download/v0.1.0-candidate/paper.pdf repository: https://github.com/ipitchford/sharp-four-factor-cohen-macaulay-obstruction archive: https://zenodo.org/records/22304827 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A sharp four-factor Cohen–Macaulay obstruction to integral Schubert deformation ## Summary An AIM problem asks whether a Cohen–Macaulay union of Schubert varieties must be able to deform to an irreducible variety. This anonymous, unrefereed candidate gives a sharp negative answer for coordinate surfaces in products of projective lines. Represent a union by a graph $G$. An edge $ij$ selects the surface on which the $i$th and $j$th coordinates can vary and all other coordinates are fixed. The candidate proves: - the union is Cohen–Macaulay exactly when the active graph is connected; - it deforms inside the same product to a geometrically integral surface exactly when the graph is complete multipartite. These two conditions first separate for the path $$ 1\mathbin{-}2\mathbin{-}3\mathbin{-}4. $$ Its three edges give three Schubert surfaces in $(\mathbf P^1)^4$. Their union is reduced, pure and Cohen–Macaulay, but has no integral deformation of the stated kind. Four active factors and three components are both minimal. ## Summary for specialists Fix $0=[1:0]$ in each factor of $P_n=(\mathbf P^1_k)^n$, where $k$ is algebraically closed. For a nonempty simple graph $G$, let $$ Y_G=\bigcup_{ij\in E(G)}X_{\{i,j\}}, $$ where $X_{\{i,j\}}$ is the coordinate Schubert surface on which precisely coordinates $i$ and $j$ vary. Connectivity is evaluated on the active vertices; inactive factors become matroid loops. The theorem candidate states that $Y_G$ is Cohen–Macaulay if and only if $G$ is connected, while $Y_G$ has an embedded flat projective deformation with geometrically integral generic fibre if and only if $G$ is complete multipartite. Both statements are characteristic-free. For $G=P_4$ with edge set $\{12,23,34\}$, basis exchange fails between $B_1=\{1,2\}$ and $B_2=\{3,4\}$. The absent replacements $\{1,3\}$ and $\{1,4\}$ obstruct an integral generic fibre. Connected graphs on at most three active vertices, and connected graphs with fewer than three edges, are complete multipartite, proving both minimality statements. ## Technical account Give the $i$th factor coordinates $[x_i:y_i]$, with $y_i=0$ at the fixed Schubert point. The Cox ring is $$ S=k[x_1,y_1,\ldots,x_n,y_n], $$ and the union has ideal $$ I(Y_G)=I_GS+(y_\ell:\ell\text{ inactive}), $$ where $I_G$ is the Stanley–Reisner ideal of the graph viewed as a pure one-dimensional simplicial complex. Thus the Cox quotient is a polynomial extension of the graph face ring. Reisner's criterion then says that it is Cohen–Macaulay exactly when the active graph is connected. The paper spells out the affine-chart Laurent-extension step connecting the Cox computation to the projective scheme. The dimension-indexed multidegrees of $Y_G$ are exactly $$ \operatorname{MSupp}(Y_G)=\{e_i+e_j:ij\in E(G)\}. $$ Multidegrees remain constant in an embedded flat projective family. The multidegree support of an integral multiprojective variety is a discrete algebraic polymatroid. Because every displayed vector is zero-one of weight two, the support must be the basis set of a rank-two matroid. After loops are deleted, the graph of two-element bases of a rank-two matroid is exactly a complete multipartite graph: its parts are the parallel classes. The converse is constructive. For the parts $C_1,\ldots,C_r$ of a complete multipartite graph, choose pairwise nonproportional vectors in $k^2$ and use the corresponding linear forms to map $\mathbf A^2$ into $(\mathbf P^1)^n$. Cross-part projections are birational and same-part projections have dimension at most one, so the closure is an integral multiplicity-free surface with precisely the desired multidegrees. Brion's multiplicity-free degeneration theorem, in the explicit scheme-theoretic form also supplied by Caminata–Cid-Ruiz–Conca, degenerates it to the exact reduced coordinate union. For the sharp path, the Cox ideal is $$ (y_3,y_4)\cap(y_1,y_4)\cap(y_1,y_2) =(y_1y_3,y_1y_4,y_2y_4). $$ Its generators are the maximal minors of an explicit $3\times2$ Hilbert–Burch matrix. The quotient has depth and dimension six, supplying a second exact check of Cohen–Macaulayness in the minimal example. ## Evidence, assurance and limitations The mathematical proof is carried by the seven-page manuscript. The immutable package adds an exact finite certificate for the four-factor ideal, Hilbert–Burch minors, basis-exchange failure and all smaller graph cases. Python replays the certificate in ordinary and optimized modes and must reject three deliberate corruptions. Macaulay2 separately recomputes the ideal, dimension, depth and projective dimension. The tagged GitHub prerelease and Zenodo record expose byte-identical copies of the PDF, complete ZIP and checksum sidecar. Public GitHub Actions checks the manifest, both Python versions, hostile controls, durable Macaulay2 record and the full ordered PDF prose stream. The release workstation additionally ran a fresh Macaulay2 replay and exact byte-for-byte PDF-text comparison. These checks do not prove the cited geometric theorems, independently validate their application, or replace the written argument. Python and Macaulay2 are producer-controlled. The supplied review, five internal role reports and one confirmation are producer-coordinated editorial evidence, not authenticated external specialist review or journal peer review. The full AIM problem remains open outside coordinate surface unions in $(\mathbf P^1)^n$. No higher-dimensional classification or result for arbitrary flag varieties is claimed. A targeted search found no exact collision, but novelty remains candidate-only and historical priority is unestablished. ## Relationship to earlier work Ardila and Boocher study closures of linear spaces in products of projective lines, with matroid-controlled multidegrees and Cohen–Macaulay initial ideals. Their dimension-two construction supplies the closest realization precedent. Their convention is complementary: codimension-indexed bases of rank $n-2$ become the rank-two bases used here after taking complements, or equivalently duals. Castillo, Cid-Ruiz, Li, Montaño and Zhang prove the general polymatroidality of multidegree support and give a twelve-factor Cohen–Macaulay support that is not a polymatroid. Brion supplies the multiplicity-free degeneration theorem, and Caminata, Cid-Ruiz and Conca give the explicit reduced coordinate-prime form used in the proof. Reisner supplies the graph face-ring criterion. The candidate contribution is the exact rank-two specialization: connectivity and complete multipartiteness become the competing graph conditions, the four-vertex path is the first obstruction, and both factor and component minimality are proved. This positioning narrows the originality claim; it does not establish priority. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Algebraic geometers | A complete coordinate-surface answer and a small obstruction to a natural converse | The full AIM classification remains open | | Commutative algebraists | A transparent Stanley–Reisner and Hilbert–Burch model | Cox-ring and projective-scheme Cohen–Macaulayness are connected through an explicit chart argument | | Matroid theorists | A sharp rank-two meeting point between basis support and multipartite graphs | Multidegree support is necessary; the converse also needs an integral realization and degeneration theorem | | Computational reviewers | Small Python and Macaulay2 checks with hostile controls | Both implementations remain producer-controlled | | Formalizers | A compact chain from graph topology to projective degeneration | Imported geometric theorems and convention changes must be formalized explicitly | | Interested readers | A four-vertex example showing why one good singularity property need not force deformability | Candidate publication is not field consensus | ## Why the problem matters Degeneration is a central way to replace a difficult variety by a combinatorial union while preserving enough information to compute. Brion's theorem says that a multiplicity-free integral variety can degenerate to a reduced Cohen–Macaulay Schubert union. The AIM question asks how much of that implication can be reversed. The surface classification identifies the missing condition exactly in one natural testbed. Connectivity controls local algebra, while matroid basis exchange controls which multidegree supports can belong to an integral generic fibre. The four-vertex path is connected but violates the second condition. That separation turns a broad deformation question into a small reusable obstruction and identifies where higher-dimensional work must go beyond graph connectivity. ## How to inspect or reproduce the recorded checks Use immutable tag `v0.1.0-candidate` or version DOI `10.5281/zenodo.22304827`, not moving `main`. ```sh python3 verify_bundle.py ``` For component diagnosis: ```sh python3 verify.py certificate.json python3 -O verify.py certificate.json python3 test_verify.py M2 --script verify.m2 ``` The composite command also checks complete manifest coverage and exact local agreement between the PDF's extracted text and `paper.txt`. The public Linux workflow uses `check_pdf_text_portable.py` because Poppler versions can change reading order inside displayed formulas; it still compares the complete ordered prose-token stream. A successful run checks package integrity and the encoded finite consequences. It does not independently prove the universal graph classification or cited geometric bridge. ## The most valuable next projects 1. Reconstruct the Cox-ring, multidegree and degeneration arguments under authenticated unaffiliated algebraic-geometry review. 2. Formalize the coordinate-surface classification and the rank-two matroid equivalence in a proof assistant. 3. Classify coordinate Schubert unions of dimension at least three, where higher-dimensional Cohen–Macaulay complexes and polymatroids replace graphs. 4. Test analogous sufficiency criteria in flag varieties beyond products of projective lines. 5. Conduct a broader specialist novelty and priority review, including institutional mathematical databases. ## What is in the evidence package The ZIP contains the DOI-bearing PDF and source, aligned Markdown, exact certificate, Python and Macaulay2 replay, hostile controls, source and citation audits, bounded novelty report, public response matrix, internal editorial reports, licences, environment declaration and complete manifest. The private supplied review is deliberately excluded because redistribution rights were not established; its SHA-256 and an original response matrix are public. The frozen ZIP is 395,619 bytes with SHA-256 `07d8c1e2f51a229a8cb52869ed4048e529a196824a078f4d4e253f832ef90108`. The version DOI is the citation target. Any mathematical correction should be released as a versioned successor rather than silently replacing this candidate. ## Open directions for follow-up research - Obtain an authenticated unaffiliated reconstruction of the Cox-ring, matroid-support and degeneration arguments in a materially separate notation and implementation. - Formally verify the coordinate-surface classification, including the scheme-theoretic bridge from multiplicity-free multidegrees to the reduced standard coordinate union. - Classify Cohen–Macaulay coordinate Schubert unions of dimension at least three and determine when their supports are algebraic polymatroids realizable by integral varieties. - Extend or refute analogous sharp classifications for Schubert unions in flag varieties beyond products of projective lines. - Obtain authenticated specialist review and a broader MathSciNet or zbMATH assessment of novelty and historical priority. ## Research process, metrics and reusable methods This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:sharp-four-factor-cohen-macaulay-obstruction - Attempt and metric receipts: ep-attempt:sharp-four-factor-cohen-macaulay-obstruction-publication-completion: published / positive; scope publication-only; target Publish a reader-first Evidence Press release with provenance-bound fable audio, deterministic art and thumbnail, reciprocal method and work records, green composite CI, guarded zero-cost deployment and exact canonical HTML, paper.json and media readback.; active forecast 70 minutes (35-120); Fermi components Reader-first page, metadata and reciprocal operating records: 1 x 10/20/35 minutes low/central/high (One compact structural theorem with a load-bearing fixed-ambient and coordinate-surface claim boundary.); Claim-disciplined art, Open Graph image, thumbnail and fable audio: 1 x 10/15/25 minutes low/central/high (One new mathematical slug using established deterministic renderers and the house TTS profile.); Composite build, protocol seals and hosted CI: 1 x 10/20/35 minutes low/central/high (One reader-first release on the current schema with reciprocal work and method records.); Merge, guarded deployment and canonical readback: 1 x 5/15/25 minutes low/central/high (One zero-cost Cloudflare deployment with exact release, media, protocol and preservation checks.); positive-signal/closure probabilities 0.99/0.92 within 120 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 21/unknown/0/5/0/3; cycles positive/negative/inconclusive 0/0/0; falsification gates 0; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -49 minutes; ratio 0.3; inside interval false; positive-signal/target-closure Brier scores 0.0001/0.0064; missing telemetry activeHumanMinutes: No instrument captured human direction or review time inside the prospective publication-only boundary.; deduplicatedModelTokens: No active fork-aware task counter exposed exact task-local model-token usage at or after the 12:35:01Z publication boundary, so tokens are not reconstructed from conversation context.; uncachedInputTokens: The runtime does not expose an uncached-input token counter.; appended measurement corrections measurement.reworkMinutes -> metrics.outcome.reworkMinutes: Retain the immutable zero-rework snapshot and record the conservative three-minute derived-index repair total only in the terminal outcome. (reason: The first public release-candidate snapshot recorded zero before the hosted whole-corpus audio-provenance failure and its bounded derived-index repair were observed.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json - Intended aims: science - Artifact roles: research-output, evidence-assessment, method-demonstration, communication - Decision object: obstruction — An exact graph classification and minimal four-factor path obstruction separating Cohen–Macaulay coordinate Schubert surface unions from those admitting embedded integral deformation. Scope: Nonempty coordinate Schubert surface unions in fixed ambient products (P1)^n over algebraically closed fields; not higher-dimensional unions, arbitrary flag varieties, independent validation, exhaustive novelty or historical priority. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: assurance, publication, translation - Semantic bridge: explicit — The paper maps coordinate surface components to graph edges, the Cox quotient to a graph Stanley–Reisner ring, flat-family multidegrees to rank-two matroid bases, complete multipartite graphs to direct integral realizations, and multiplicity-free classes to the exact reduced coordinate special fibre. Remaining risks: The specialization of the cited geometric theorems and the Cox-chart descent have not been reconstructed by an unaffiliated algebraic geometer.; The two finite computational routes are producer-controlled and do not verify the universal geometric proof.; The targeted literature search may miss an implicit or differently phrased prior classification.; The result covers coordinate surfaces only and supplies no higher-dimensional or general-flag classification.. - Human judgement gates: Check the fixed-ambient embedded deformation definition and keep it separate from abstract smoothability or deformations in another ambient space.; Verify the dimension-indexed multidegree convention and complementary Chow exponents before applying the rank-two basis classification.; Audit the scheme-theoretic multiplicity-free degeneration bridge rather than inferring the special fibre from cycle equality alone.; Keep the complete coordinate-surface result separate from the full AIM problem in higher dimensions and other flag varieties.; Treat producer replay, internal editorial closure, external specialist review, formal verification, novelty and priority as separate assurance dimensions. - Next assurance action: Obtain authenticated unaffiliated specialist reconstruction of the Cox-ring and degeneration bridges, then formalize the complete coordinate-surface classification and investigate the first higher-dimensional case. - Claim ceiling: An anonymous, AI-assisted, unrefereed theorem candidate classifying Cohen–Macaulayness and embedded integral deformability for coordinate Schubert surface unions in (P1)^n, with a sharp four-factor path obstruction, complete written proof, public immutable assets, producer replay and internal editorial closure; not the full AIM problem, independent validation, formal verification, authenticated external specialist review, editorial peer review, exhaustive novelty, historical priority or demonstrated workflow impact. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Faster or more reliable classification and assurance of Schubert-degeneration obstructions in AI-assisted structural reduction, exact finite replay, internal review and guarded candidate publication; design none; comparator No matched conventional research, specialist-review or publication workflow was registered.; estimand No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, citation or field outcomes was estimated.; no real-world effect evidence asserted - Parent handoffs: depends-on-claim https://aimath.org/pastworkshops/degenalggeomproblems.pdf; inherited claim: AIM Problem 2.2 asks which subsets of Schubert varieties deform to irreducibles, whether Cohen–Macaulayness is sufficient, and identifies (P1)^n as the first case to try.; inherited ceiling: The problem source fixes the broader question but does not state or establish this coordinate-surface classification, four-factor obstruction, correctness, novelty or priority.; extends-result https://doi.org/10.1007/s10801-015-0634-x; inherited claim: Ardila and Boocher construct closures of linear spaces in products of lines with matroid-controlled multidegrees and Cohen–Macaulay initial ideals.; inherited ceiling: Their construction supplies the adjacent integral realization and dual matroid convention, not the exact located connectivity versus complete-multipartiteness statement or sharp path minimality.; extends-result https://doi.org/10.1016/j.aim.2020.107382; inherited claim: Castillo and collaborators prove that multidegree support of an integral multiprojective variety is a discrete algebraic polymatroid and exhibit a larger Cohen–Macaulay non-polymatroidal support.; inherited ceiling: Their general obstruction supplies the support mechanism, not the rank-two graph classification, four-factor witness or minimality theorem. ## Verification status Anonymous, AI-assisted, unrefereed algebraic-geometry theorem candidate after actioning a supplied Minor Revision review, completing a five-role producer-coordinated editorial gate and passing one frozen-target confirmation with no P0 or P1 findings. The complete coordinate-surface classification is written as a mathematical proof and supported by finite producer-side replay. The result does not classify higher-dimensional Schubert unions or arbitrary flag varieties. Novelty is candidate-only and priority is unestablished. ## References 1. American Institute of Mathematics. Degenerations in Algebraic Geometry: Problem Session, Problem 2.2 (2016). 2. Ardila, F., and Boocher, A. (2016). The closure of a linear space in a product of lines. Journal of Algebraic Combinatorics 43, 199–235. 3. Brion, M. (2003). Multiplicity-free subvarieties of flag varieties. Contemporary Mathematics 331, 13–23. 4. Castillo, F., Cid-Ruiz, Y., Li, B., Montaño, J., and Zhang, N. (2020). When are multidegrees positive? Advances in Mathematics 374, 107382. 5. Caminata, A., Cid-Ruiz, Y., and Conca, A. (2023). Multidegrees, prime ideals, and non-standard gradings. Advances in Mathematics 435, 109361. 6. Reisner, G. A. (1976). Cohen–Macaulay quotients of polynomial rings. Advances in Mathematics 21, 30–49.