Sequences of monoidal transformations of a regular noetherian local domain

Abstract:

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