Directed unions of local quadratic transforms of a regular local ring

Abstract 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 { ( R n , m n ) } n ≥ 0 of d -dimensional regular local rings dominated by V with R = R 0 and R n + 1 the local quadratic transform of R n along V . Let S : = ⋃ n ≥ 0 R n . 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 { ( R n , m n ) } n ≥ 0 and the integral domain S = ⋃ n ≥ 0 R n may be defined without first prescribing a dominating valuation domain V . If { ( R n , m n ) } 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 ) = lim n → ∞ ord R n ( f ) ord R n ( g ) . If { ( R n , m n ) } n ≥ 0 is a family of monomial local quadratic transforms, we give necessary and sufficient conditions for { ( R n , m n ) } 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 ( m i ) . 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.