Sequences of monoidal transformations of a regular noetherian local domain

Let RR be a regular noetherian local ring of dimension d≥2d\geq 2. We characterize the sequences (Ri)i≥0(R_i)_{i\geq 0} of successive monoidal transforms of R=R0R=R_0 such that S=⋃i≥0RiS =\bigcup _{i\geq 0} R_i is a valuation ring. This characterization involves two well-known conditions in the case of quadratic transforms ((Ri)i≥0(R_i)_{i\geq 0} either switches strongly infinitely often or is height one directed), to which we must add the condition that a family of ideals of SS (finitely supported on the exceptional divisors along the sequence) is linearly ordered by inclusion. Moreover and under the assumption that SS is a valuation ring, we compute the limit points (in the Zariski–Riemann space over the quotient field of RR equipped with the patch topology) of the valuation rings associated with the order valuations defined by the centers of the monoidal transforms as well as the limit points of the valuation rings associated with the order valuations defined by the maximal ideals of the rings RiR_i, i≥0i\geq 0.