Asymptotic properties of infinite directed unions of local quadratic transforms

Abstract

Let (R,m) be a regular local ring of dimension at least 2. For each valuation domain birationally dominating R, there is an associated sequence {Rn} of local quadratic transforms of R. We consider the case where this sequence {Rn}n≥0 is infinite and examine properties of the integrally closed local domain S=⋃n≥0Rn in the case where S is not a valuation domain. For this sequence, there is an associated boundary valuation ring V=⋃n≥0⋂i≥nVi, where Vi is the order valuation ring of Ri. There exists a unique minimal proper Noetherian overring T of S. T is the regular Noetherian UFD obtained by localizing outside the maximal ideal of S and S=V∩T. In the present paper, we define functions w and e, where w is the asymptotic limit of the order valuations and e is the limit of the orders of transforms of principal ideals. We describe V explicitly in terms of w and e and prove that V is either rank 1 or rank 2. We define an invariant τ associated to S that is either a positive real number or +∞. If τ is finite, then S is archimedean and T is not local. In this case, the function w defines the rank 1 valuation overring W of V and W dominates S. The rational dependence of τ over w(T×) determines whether S is completely integrally closed and whether V has rank 1. We give examples where S is completely integrally closed. If τ is infinite, then S is non-archimedean and T is local. In this case, the function e defines the rank 1 valuation overring E of V. The valuation ring E is a DVR and E dominates T, and in certain cases we prove that E is the order valuation ring of T.