Directed unions of local quadratic transforms of a regular local ring

Let (R,m) be a d-dimensional regular local domain with d≥2 and let V be a valuation domain birationally dominating R such that the residue field of V is algebraic over R/m. Let v be a valuation associated to V. Associated to R and V there exists an infinite directed family {(Rn,mn)}n≥0 of d-dimensional regular local rings dominated by V with R=R0 and Rn+1 the local quadratic transform of Rn along V. Let S:=⋃n≥0Rn. Abhyankar proves that S=V if d=2. Shannon observes that often S is properly contained in V if d≥3, and Granja gives necessary and sufficient conditions for S to be equal to V. The directed family {(Rn,mn)}n≥0 and the integral domain S=⋃n≥0Rn may be defined without first prescribing a dominating valuation domain V. If {(Rn,mn)}n≥0 switches strongly infinitely often, then S=V is a rank one valuation domain and for nonzero elements f and g in m, we have v(f)v(g)=limn→∞ordRn(f)ordRn(g). If {(Rn,mn)}n≥0 is a family of monomial local quadratic transforms, we give necessary and sufficient conditions for {(Rn,mn)}n≥0 to switch strongly infinitely often. If these conditions hold, then S=V is a rank one valuation domain of rational rank d and v is a monomial valuation. Assume that V is rank one and birationally dominates S. Let s=∑i=0∞v(mi). Granja, Martinez and Rodriguez show that s=∞ implies S=V. We prove that s is finite if V has rational rank at least 2. In the case where V has maximal rational rank, we give a sharp upper bound for s and show that s attains this bound if and only if the sequence switches strongly infinitely often.