Residue ideals of hyperplane arrangements

In this paper, we introduce a new idea to study modules of logarithmic differential forms of hyperplane arrangements, which we call residue ideals. We first establish basic properties of these ideals, including their radicals and primary decompositions, and obtain applications for freeness of restrictions of arrangements. Then we apply these ideals to the study of modules of logarithmic differential $1$-forms for graphic arrangements. We give an explicit generating set for these modules and find a new connection to cover ideals of graphs studied in combinatorial commutative algebra. As a consequence we establish several new connections between arrangement theory and Stanley--Reisner theory.