The normal reduction number of two-dimensional cone-like singularities
Let $(A, \mathfrak m)$ be a normal two-dimensional local ring and $I$ an $\mathfrak m$-primary integrally closed ideal with a minimal reduction $Q$. Then we calculate the numbers: $\mathrm{nr}(I) = \min\{n \;|\; \overline{I^{n+1}} = Q\overline{I^n}\}, \quad \bar{r}(I) = \min\{n \;|\; \overline{I^{N+1}} = Q\overline{I^N}, \forall N\ge n\}$, $\mathrm{nr}(A)$, and $\bar{r}(A)$, where $\mathrm{nr}(A)$ (resp. $\bar{r}(A)$) is the maximum of $\mathrm{nr}(I)$ (resp. $\bar{r}(I)$) for all $\mathfrak m$-primary integrally closed ideals $I\subset A$. Then we have that $\bar{r}(A) \le p_g(A) + 1$, where $p_g(A)$ is the geometric genus of $A$. In this paper, we give an upper bound of $\bar{r}(A)$ when $A$ is a cone-like singularity (which has a minimal resolution whose exceptional set is a single smooth curve) and show, in particular, if $A$ is a hypersurface singularity defined by a homogeneous polynomial of degree $d$, then $\bar{r}(A)= \mathrm{nr}(\mathfrak m) = d-1$. Also we give an example of $A$ and $I$ so that $\mathrm{nr}(I) = 1$ but $\bar{r}(I)= \bar{r}(A) = p_g(A) +1=g+1$ for every integer $g \ge 2$.