Semigroups of valuations on local rings, II

Given a noetherian local domain $R$ and a valuation $\nu$ of its field of fractions which is nonnegative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values lying in certain parts of the value group $\Gamma$ of $\nu$. We show that this growth condition imposes restrictions on the semigroups $\nu(R\setminus \{0\})$ for noetherian $R$ which are stronger than those resulting from the previous paper of the first author. Given an ordered embedding $\Gamma\subset ({{\bf R}}^h)_{\hbox{\rm lex}}$, where $h$ is the rank of $\nu$, we also study the shape in ${{\bf R}}^h$ of the parts of $\Gamma$ which appear naturally in this study. We give examples which show that this shape can be quite wild in a way which does not depend on the embedding and suggest that it is a good indicator of the complexity of the semigroup $\nu(R\setminus \{0\})$.