Characteristic variety of the Gauss-Manin differential equations of a generic parallelly translated arrangement

We consider a weighted family of $n$ generic parallelly translated hyperplanes in ${\text{C}}^k$ and describe the characteristic variety of the Gauss-Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the Plucker coordinates of the associated point in the Grassmannian Gr(k,n). The Laurent polynomials are in involution.