Growth of rank 1 valuation semigroups

Abstract

We consider the question of which semigroups can occur as the semigroup $S_R(\nu)$ of positive values of a rank 1 valuation dominating a Noetherian local ring $R$. We give a number of bounds of polynomial type on the growth of $\phi(n)=S_R(\nu)\cap (0,n)$ for $n\in\NN$, starting with the upper bound of $P_R(n)$, where $P_R(n)$ is the Hilbert function of $R$. This bound is generalized to an extremely general bound for arbitrary rank valuations in the paper "Semigroups of valuations on local rings, II", by Cutkosky and Teissier, arXiv:0805.3788. This bound is already enough to give simple examples of rank 1 well ordered semigroups which are not the value semigroup $S_R(\nu)$ of a valuation dominating a Noetherian local ring. In the case of rank 1, it is possible to give more precise estimates of $\phi(n)$, which we prove in this paper. We also give examples showing that many different rates of growth are possible for $\phi(n)$ on a regular local ring of dimension 2, such as $n({\alpha}$ for any rational $\alpha$ with $1\le\alpha\le 2$, and $n{log}(n)$.