Overrings of two-dimensional Noetherian domains representable by Noetherian spaces of valuation rings

Let D be a Noetherian domain of Krull dimension 2, and let Hsubset of or equal toR be integrally closed overrings of D. We examine when H can be represented in the form H=(intersection operatorVset membership, variantΣV)∩R, with Σ a Noetherian subspace of the Zariski–Riemann space of the quotient field of D. We characterize also the special case in which Σ can be chosen to be a finite character collection of valuation overrings of D.