Tangent classes of matroids and wonderful compactifications

Abstract: For every loopless matroid $M$ and every Feichtner--Yuzvinsky building set $\mathcal{G}$ containing the top flat, we construct an integral tangent class $$T_{M,\mathcal{G}}^{{\mathbb{Z}}\in} K_{\mathbb{Z}}(M,\mathcal{G})$$; in the realizable case it specializes to the class of the tangent bundle of the corresponding wonderful compactification, it recovers the Hilbert series of the Chow ring through Hirzebruch--Riemann--Roch, and it satisfies the expected Chern-alpha lower bounds. This reproduces the tangent class and its key properties studied by the first author in arXiv:2606.22650. The main body of this paper was produced autonomously, without human mathematical guidance, by Danus, an AI mathematical reasoning agent. Danus solved the problem before arXiv:2606.22650 was publicly available, demonstrating the potential of AI agents in mathematical research. We reproduce its output faithfully, adding only editorial comments; the experiment is documented in Appendix B.

• The paper introduces intrinsic integral tangent classes for matroids, providing explicit constructions and rigorous verification of key properties.
• It employs functorial descent and one-flat refinement techniques to relate combinatorial K-theory with tangent bundles of wonderful compactifications.
• The results establish sharp Chern-number lower bounds and complete integrality, bridging combinatorial and geometric invariants in matroid theory.

Tangent Classes of Matroids and Wonderful Compactifications

Overview and Objectives

This paper introduces the construction and analysis of intrinsic integral tangent classes for matroids and their associated wonderful compactifications, specifically for all loopless matroids and any Feichtner–Yuzvinsky building set containing the top flat. The authors develop an integral class $T_{M,\mathcal{G}}$ in the combinatorial K-theory ring $K_{\mathbb{Z}}(M, \mathcal{G})$, provide a full verification of its core properties—including comparison with tangent bundles of wonderful compactifications, recovery of the Hilbert series, and Chern-number lower bounds—and give a rigorous account of its integral and rational structure.

Notably, the bulk of the main mathematical results were produced by Danus, an autonomous reasoning agent, prior to the public dissemination of closely related human work, thus providing both new mathematical results and a demonstration of AI systems for extended open-problem research in algebraic geometry and combinatorics.

Construction of the Integral Tangent Class

For a loopless matroid $M$ of rank $d+1$ and a suitable building set $\mathcal{G}$, the paper defines the combinatorial integral K-ring $K_{\mathbb{Z}}(M, \mathcal{G})$ generated by classes $\tau_F$ for $F \in \mathcal{G}$, subject to relations encoding nestedness and atom (rank-one flat) conditions. The main object is the integral tangent class

$$T_{M,\mathcal{G}} \coloneqq \sum_{F \in \mathcal{G}^\circ} (1 - \tau_F)^{-1} - Q_{M,\mathcal{G}}^{\mathbb{Z}}$$

where $Q_{M,\mathcal{G}}^{\mathbb{Z}}$ is an explicit integral representative descending from the Berget–Eur–Spink–Tseng (BEST) tautological quotient class.

Key steps in the construction include:

• Descent from the maximal model via one-flat refinement chains in building sets, utilizing properties of the K-theory and Chow rings;
• Rationalization and comparison with known rational classes, ensuring that the integral class agrees rationally with the canonical tangent class;
• Functoriality and compatibility with the realizable case, wherein $K_{\mathbb{Z}}(M, \mathcal{G})$0 arises from a linear subspace and wonderful compactification geometry applies.

Theoretical Properties and Main Results

The main theorem establishes that the constructed $K_{\mathbb{Z}}(M, \mathcal{G})$1 satisfies three core properties, for all loopless matroids and all top-containing building sets:

1. Compatibility with Wonderful Compactifications: If $K_{\mathbb{Z}}(M, \mathcal{G})$2 is realizable as a linear space $K_{\mathbb{Z}}(M, \mathcal{G})$3 and $K_{\mathbb{Z}}(M, \mathcal{G})$4 is the corresponding wonderful compactification, there is an explicit isomorphism of $K_{\mathbb{Z}}(M, \mathcal{G})$5-rings taking $K_{\mathbb{Z}}(M, \mathcal{G})$6 to the tangent bundle class $K_{\mathbb{Z}}(M, \mathcal{G})$7 in $K_{\mathbb{Z}}(M, \mathcal{G})$8.
2. Hirzebruch–Riemann–Roch (Hilbert Identity): The K-theoretic Todd polynomial of $K_{\mathbb{Z}}(M, \mathcal{G})$9 exactly recovers the Hilbert series of the matroid Chow ring:

$M$0

Moreover, this equates dimensions of graded pieces of the Chow ring with formal Hirzebruch–Riemann–Roch invariants of exterior powers of the dual tangent class.

1. Chern-$M$1 Lower Bounds: For each $M$2, there is an explicit combinatorial lower bound:

$M$3

where $M$4 is the negative of the top-flat generator.

These results hold integrally in $M$5 for all building sets, including those whose associated nested fans are not complete, so the construction is truly intrinsic and independent of global toric geometry.

Descent, Functoriality, and One-Flat Techniques

A novel technical contribution is the development of a saturated, functorial descent machinery for both the integral quotient class and the tangent class, propagated along one-flat refinements of building sets. This involves a detailed analysis of filtration, associated gradeds, and the interplay with the atom linear relations, as well as comparison morphisms between K-rings and their Chow counterparts.

Integral surjectivity and saturation at each stage are established via Smith normal form and detailed tracking of the exceptional locus in blowup diagrams, ensuring that the entire construction proceeds integrally, not merely rationally.

Strong Numerical and Structural Results

The following strong claims are rigorously established and highlighted in the paper:

• Integral Structure: $M$6 has integer coefficients in the standard monomial basis and is well-defined in all cases, independent of completeness assumptions on associated fans.
• Realizable/Geometric Specialization: The intrinsic $M$7-ring isomorphism specializes (when realizable) to the explicit geometric identification with the $M$8-theory ring of the wonderful compactification, matching generators and the tangent bundle class.
• Equality of Invariants: The Hirzebruch–Riemann–Roch formula precisely recovers the Hilbert function, showing that the combinatorial and geometric invariants coincide for all matroids and building sets.
• Chern Number Bounds: The lower bound $M$9 for generalized Chern numbers is achieved intrinsically, generalizing previous conjectures and results for special cases.

These claims are fully verified for all values in the valid range. The only minor gap is an incomplete proof for an auxiliary lemma (truncation-transfer for the Chern-alpha bound), which is externally known to be true and referenced to human literature.

Implications and Future Directions

The results have several significant theoretical and practical implications:

• Universality: The intrinsic construction applies to all (possibly non-realizable) matroids and all Feichtner–Yuzvinsky building sets, providing a unified approach that generalizes and extends geometric, combinatorial, and toric models.
• K-theoretic and Cohomological Computations: The explicit formulae for the tangent class and Chern numbers yield new tools for studying intersection theory, positivity, and $d+1$0-ring structures in matroid contexts.
• Foundations for Further AI-driven Mathematical Research: By directly demonstrating the ability of autonomous systems to resolve extended open mathematical problems and to write complete mathematical manuscripts with correct technical content, the methodology paves the way for scaled deployments in other branches of pure mathematics.

Extensions could include detailed analyses of torsion phenomena in the integral K-rings, explorations of alternative building set combinatorics, further integration with equivariant and motivic invariants, and broadening AI-driven research to additional open problems in algebraic combinatorics and related fields.

Conclusion

This work rigorously establishes the existence and properties of intrinsic integral tangent classes for matroids and their wonderful compactifications, unifying combinatorial K-theory, intersection theory, and geometric models. The construction, verified mathematically in both integral and rational settings and realized by autonomous reasoning, substantiates the equality between algebraic and combinatorial invariants for a wide class of objects, and supplies new tools and perspectives for the study of matroid invariants and their applications.