Norms as products of linear polynomials

Let F be a number field, and let F\subset K be a field extension of degree n. Suppose that we are given 2r sufficiently general linear polynomials in r variables over F. Let X be the variety over F such that the F-points of X bijectively correspond to the representations of the product of these polynomials by a norm from K to F. Combining the circle method with descent we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X.