Complete intersection points on affine varieties

This paper addresses the following problem: given a commutative ring A, what is the structure of the set of “CI points,” i.e., those maximal ideals generated by dim(A) elements? When A is finitely generated over an algebraically closed field, we conjecture that this set is a countable union of closed subsets of Max(A). When A is regular of dimension ? or 3, we verify this conjecture, as well as an analogous set-theoretic conjecture.