Geometric Arveson-Douglas conjecture for the Drury-Arveson space: The case of one-dimensional variety

We consider a class of analytic subsets M˜ of an open neighborhood of the closed unit ball in Cⁿ. Such an M˜ gives rise to a submodule R and a quotient module Q of the Drury-Arveson module H_n² in n variables. The geometric Arveson-Douglas conjecture predicts that the quotient module Q is p-essentially normal for p>d=dim_CM˜. We prove this conjecture for the case of dimension d=1. In fact, we prove that if d=1, then Q is 1-essentially normal, which is a stronger result than the original prediction.