AI 中文总结
该研究定义了图构形空间的边-陈代数，证明其为图不变量，在三维环境下可严格细化色多项式与Tutte多项式，对树给出闭式希尔伯特级数公式，且其三次关系数据可作为树的完全不变量。
AI 中文摘要
将有限图的顶点置于射影点上，每条边定义一个到Gr(2,n)的张映射，拉回的陈类生成一个分次代数A_G^(n)。该赋值是协变图函子，因此抽象分次代数类型是图不变量。在四维环境下，其希尔伯特级数与色多项式、Tutte多项式不可比；在三维环境下的5顶点图中，34类图对应33种代数类型，严格细化了这两类经典多项式。对每棵树T，可得到仅依赖|V(T)|和n的闭式希尔伯特级数公式，且A_T^(3)可唯一确定T的同构类，更精确地说，其三次关系数据是多项式规模的完全树不变量。最后，图构形空间作为稠密开集嵌入图片簇，且具有相同加性同调的图片簇可拥有非同构的边-陈代数。
英文摘要
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.
Comments24 pages, no figures. Accompanying software and computational certificates: https://doi.org/10.5281/zenodo.21538969
