Press release · 4 September 2026 · version 0.1.0-candidate
A sharp four-factor Cohen–Macaulay obstruction to integral Schubert deformation
Coordinate Schubert surface unions are Cohen–Macaulay exactly for connected active graphs, but deform to integral surfaces exactly for complete multipartite graphs.
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
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
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
and the union has ideal
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
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
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.
python3 verify_bundle.pyFor component diagnosis:
python3 verify.py certificate.json
python3 -O verify.py certificate.json
python3 test_verify.py
M2 --script verify.m2The 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
- Reconstruct the Cox-ring, multidegree and degeneration arguments under authenticated unaffiliated algebraic-geometry review.
- Formalize the coordinate-surface classification and the rank-two matroid equivalence in a proof assistant.
- Classify coordinate Schubert unions of dimension at least three, where higher-dimensional Cohen–Macaulay complexes and polymatroids replace graphs.
- Test analogous sufficiency criteria in flag varieties beyond products of projective lines.
- 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.
Media
The audio briefing is provided in the header above. Download the MP3 briefing · read the transcript.
Open directions for follow-up research
Also available in machine-readable form for research agents and follow-up projects.
- 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.
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.
Cite
BibTeX
@misc{sharpfourfactorcohenmacaulayobstruction2026,
title = {A sharp four-factor Cohen–Macaulay obstruction to integral Schubert deformation},
author = {Anonymous},
year = {2026},
doi = {10.5281/zenodo.22304827},
url = {https://doi.org/10.5281/zenodo.22304827},
version = {0.1.0-candidate},
howpublished = {Zenodo},
note = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/sharp-four-factor-cohen-macaulay-obstruction/}
}Also: cite.bib · paper.json · this page as Markdown