Sequences of quadratic transforms of a noetherian local domain

Abstract

Let R be a noetherian local domain. The aim of this paper is to characterize the sequences (Ri)i≥0 of successive quadratic transforms of R=R0 such that S=⋃i≥0Ri is a valuation ring. These characterizations involve ideals of S supported on the exceptional divisors along the sequence. Furthermore, the characterizations are improved remarkably when the Hilbert functions of the rings (Ri)i≥0 stabilize along the sequence. We also give some examples to put our results in perspective.