Integrally closed rings in birational extensions of two-dimensional regular local rings

Abstract: Let $D$ be an integrally closed local Noetherian domain of Krull dimension 2, and let $f$ be a nonzero element of $D$ such that $fD$ has prime radical. We consider when an integrally closed ring $H$ between $D$ and $D_f$ is determined locally by finitely many valuation overrings of $D$. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of $D$, and, when $D$ is analytically normal, this property holds for $D$ if and only if it holds for the completion of $D$. This latter fact, along with MacLane's notion of key polynomials, allows us to prove that in some central cases where $D$ is a regular local ring and $f$ is a regular parameter of $D$, then $H$ is determined locally by a single valuation. As a consequence, we show that if $H$ is also the integral closure of a finitely generated $D$-algebra, then the exceptional prime ideals of the extension $H/D$ are comaximal. Geometrically, this translates into a statement about intersections of irreducible components in the closed fiber of the normalization of a proper birational morphism.