Nevanlinna-Pick Families and Singular Rational Varieties

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 $A \subset \mathbb{C}[z_1,\ldots,z_d]$, such that the map from the affine space to the spectrum of $A$ is an isomorphism except for finitely many points. Letting $\mathfrak{A}$ be the weak-$*$ closure of $A$ in $\mathcal{M}_d$ -- the multiplier algebra of the Drury-Arveson space. We provide a parametrization for the Nevanlinna-Pick family of $M_k(\mathfrak{A})$ for $k \geq 1$. In particular, when $k=1$ the parameter space for the Nevanlinna-Pick family is the Picard group of $A$.