We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterization of the distributions satisfying the original CI statements. Classically, these ideals are generated by 2-minors of a matrix of variables; however, in the presence of hidden variables, they contain higher degree minors. This leads to the study of the structure of determinantal hypergraph ideals whose decompositions can be understood in terms of point and line configurations in the projective space.

我们研究了满足特定条件独立性 (CI) 约束的概率分布与入射几何中的点线排列之间的联系。对于 CI 语句族，我们将其代数不变量编码在超图中的多项式理想关联起来。这些理想的主要分解给出了满足原始 CI 陈述的分布的特征。经典地，这些理想是由变量矩阵的 2-minor 生成的；然而，在存在隐藏变量的情况下，它们包含更高程度的未成年人。这导致了对行列式超图理想结构的研究，其分解可以根据射影空间中的点和线配置来理解。