The goal of this note is to apply ideas from commutative algebra (a.k.a. affine algebraic geometry) to the question of constrained Nevanlinna-Pick interpolation. More precisely, we consider subalgebras A⊂ ℂ[ z1, …, zd], such that the map from the affine space to the spectrum of A is an isomorphism except for finitely many points. Letting be the weak-∗ closure of A in ℳd —the multiplier algebra of the Drury-Arveson space. We provide a parametrization for the Nevanlinna-Pick family of Mk for k ≥ 1. In particular, when k = 1 the parameter space for the Nevanlinna-Pick family is the Picard group of A.