Iterated blowups of two-dimensional regular local rings

Abstract

We explore sequences of iterated blowups of two-dimensional regular local rings. Classical results of Zariski and Abhyankar show that the directed union of blowups of this type is a valuation ring.We show that the value group of such a valuation ring is determined by the irrational number $\gamma$ that is the value of an infinite continued fraction associated to the sequence of blowups.