Edge-Span Chern Algebras of Graphical Configuration Spaces

Abstract

Place the vertices of a finite graph at projective points. Each edge defines a span map to $\mathrm{Gr}(2,n)$; the pulled-back Chern classes generate a graded algebra $A_G^{(n)}$. This assignment is a covariant graph functor, so the abstract graded-algebra type is a graph invariant. In ambient dimension four, the Hilbert series is incomparable with the chromatic and Tutte polynomials. On five vertices in dimension three, the $34$ graph classes yield $33$ algebra types, strictly refining both classical polynomials. For every tree $T$, we obtain a closed Hilbert-series formula depending only on $|V(T)|$ and $n$, while $A_T^{(3)}$ determines $T$ up to isomorphism. More precisely, its cubic relation data is a complete tree invariant of polynomial size. Finally, the graphical configuration space embeds as a dense open in the picture variety, and picture spaces with equal additive homology can have nonisomorphic edge-Chern algebras.

Disclosure

“out the collision boundary of the picture variety? Acknowledgments This disclosure follows the recommendations of the Leiden Declaration on Artificial Intelli- gence and Mathematics [2]. OpenAI’s Codex was used extensively to develop the construction and arguments, draft and revise the manuscript, conduct preliminary literature searches, and implement and test the software. Anthropic’s Claude was used for discussion and editorial f”

PDF page 23

Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T19:16:13.425727Z

Structural counts